Inclusion Criteria:

• Histologically confirmed recurrent (not amenable to curative treatment with local and/or systemic therapies) or metastatic (disseminated) HNSCC of the oral cavity, oropharynx, hypopharynx, and/or larynx that is considered incurable by local therapies.
• Disease progression at any time during or after treatment with a platinum-containing (e.g., carboplatin or cisplatin) regimen.
• Disease progression on or after treatment with an anti-PD-1/PD-L1 mAb (programmed cell death protein 1/programmed death-ligand 1 monoclonal antibody).
• Pre-study imaging that demonstrates evidence of disease progression based on investigator review of at least 2 pre-study images per RECIST 1.1, following initiation of treatment with a PD-1/PD-L1 inhibitor.
• Measurable disease by CT or MRI based on Response Criteria in Solid Tumors Version 1.1 (RECIST 1.1) as verified by blinded independent central review (BICR). Tumor lesions situated in a previously irradiated area are considered measurable if progression has been demonstrated in such lesions.
• ECOG performance status of 0 or 1 assessed within 7 days of the first dose of study intervention.
• Male participants are eligible to participate if they agree to the following during the intervention period and for at least 1 week after the last dose of lenvatinib, 3 months after the last dose of capecitabine and paclitaxel, and and 6 months after the last dose of docetaxel:
• Refrain from donating sperm
• Be abstinent from heterosexual intercourse as their preferred and usual lifestyle (abstinent on a long term and persistent basis) and agree to remain abstinent; or must agree to use contraception unless confirmed to be azoospermic.
• Contraceptive use by men should be consistent with local regulations regarding the methods of contraception for those participating in clinical studies.
• A female participant is eligible to participate if she is not pregnant or breastfeeding, and at least one of the following conditions applies:
• Is not a woman of childbearing potential (WOCBP)
• Is a WOCBP and using a contraceptive method that is highly effective (with a failure rate of <1% per year), with low user dependency or be abstinent from heterosexual intercourse as their preferred and usual lifestyle (abstinent on a long term and persistent basis), during the intervention period and for at least 120 days post pembrolizumab or 1 month post lenvatinib, whichever occurs last (Arms 1 and 3), or during the intervention period and for at least 6 months after the last dose of capecitabine, docetaxel, paclitaxel; and 2 months after the last dose of cetuximab (Arm 2).
• Female participants who randomize to Arm 2 must also agree not to donate or freeze/store eggs during the intervention period and for at least 6 months after the last dose of capecitabine, docetaxel, paclitaxel; and 2 months after the last dose of cetuximab.
• Contraceptive use by women should be consistent with local regulations regarding the methods of contraception for those participating in clinical studies.
• Adequately controlled blood pressure (BP) with or without antihypertensive medications.
• Adequate organ function.

Exclusion Criteria:

• Carcinoma of the nasopharynx, salivary gland, unknown primary origin, or nonsquamous histologies as primary tumors.
• Disease that is suitable for local therapy administered with curative intent.
• Life expectancy of less than 3 months and/or has rapidly progressing disease in the opinion of the treating investigator.
• History of (noninfectious) pneumonitis/interstitial lung disease that required steroids, or has current pneumonitis/interstitial lung disease.
• Active infection requiring systemic therapy.
• Diagnosis of immunodeficiency or is receiving chronic systemic steroid therapy (in dosing exceeding 10 mg daily of prednisone equivalent) or any other form of immunosuppressive therapy within 7 days prior to the first dose of study drug.
• Known active central nervous system (CNS) metastases and/or carcinomatous meningitis.
• Known additional malignancy that is progressing or has required active systemic treatment within the past 3 years, except basal cell carcinoma of the skin, squamous cell carcinoma of the skin, superficial bladder cancer, or carcinoma in situ that have undergone potentially curative therapy.
• Active autoimmune disease that has required systemic treatment in the past 2 years.
• Had an allogeneic tissue/solid organ transplant.
• Known history of human immunodeficiency virus (HIV) infection.
• History of hepatitis B or known active hepatitis C virus.
• History of any contraindication or has a severe hypersensitivity to any components of pembrolizumab, lenvatinib or SOC chemotherapy.
• Pre-existing ≥Grade 3 gastrointestinal or non-gastrointestinal fistula.
• History of a gastrointestinal malabsorption or any other condition or procedure that may affect oral study drug absorption.
• Had major surgery within 3 weeks prior to first dose of study interventions.
• Clinically significant cardiovascular impairment within 12 months of the first dose of study drug.
• Active tuberculosis.
• Has difficulty swallowing capsules or ingesting a suspension orally, or by a feeding tube.
• Prior treatment with lenvatinib.
• Prior chemotherapy, targeted small molecule therapy, or radiation therapy within 2 weeks prior to Study Day 1 or has not recovered from adverse events (AEs) due to a previously administered agent.
• Has received a live or live attenuated vaccine within 30 days prior to the first dose of study intervention. Note: Administration of killed vaccines is allowed.
• Previously treated with 4 or more systemic regimens given for recurrent/metastatic disease.
• Currently participating in or has participated in a study of an investigational agent and received study therapy or used an investigational device within 4 weeks prior to the first dose of study intervention.
• Known psychiatric or substance abuse disorder that would interfere with the participant's ability to cooperate with the requirements of the study.

Inclusion Criteria:

• Has histologically confirmed new diagnosis of resectable, non-metastatic, squamous cell carcinoma that is either: Stage III Human Papillomavirus (HPV) positive oropharyngeal primary that is tumor size (T) 4, lymph node involvement (N) 0-2, no distant metastases (M0); Stage III or IVA oropharyngeal HPV negative; or Stage III or IVA larynx/hypopharynx/oral cavity primaries
• Is eligible for primary surgery based on investigator decision and per local practice
• Female and male participants of reproductive potential must agree to use adequate contraception throughout the study period and for up to 180 days after the last dose of study therapy
• Male participants must refrain from donating sperm throughout the study period and for up to 180 days after the last dose of study therapy
• Female participant that is not pregnant or breastfeeding
• Has evaluable tumor burden (measurable and/or non-measurable tumor lesions) assessed by computed tomography (CT) scan or magnetic resonance imaging (MRI), based on RECIST version 1.1
• Has provided newly obtained core or excisional biopsy of a tumor lesion not previously irradiated
• Has results from testing of HPV status for oropharyngeal cancer defined as p16
• Has Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 performed within 10 days of randomization

Exclusion Criteria:

• Has Stage T4B and/or N3 LA HNSCC and/or distant metastases
• Has cancer outside of the oropharynx, larynx, and hypopharynx or oral cavity, such as nasopharyngeal, sinus, other para-nasal, or other unknown primary head and neck cancer (HNC)
• Female participant who has a positive urine pregnancy test within 72 hours prior to study start or within 24 hours prior to the start of radiotherapy with or without cisplatin
• Has received prior therapy with an anti-programmed cell death receptor 1(PD-1), anti-programmed cell death receptor ligand 1(PD-L1), or anti-programmed cell death receptor ligand 2 (PD-L2) agent or with an agent directed to another co-inhibitory T-cell receptor
• Has received prior radiotherapy treatment or systemic anti-cancer therapy including investigational agents for the HNC under study prior to study start
• Has received a live vaccine within 30 days prior to randomization
• Is currently participating in or has participated in a study of an investigational agent or has used an investigational device within 4 weeks prior to randomization
• Has a diagnosis of immunodeficiency or is receiving systemic steroid therapy (in dosing exceeding 10 mg daily of prednisone equivalent) or any other form of immunosuppressive therapy within 7 days prior to randomization
• Has a known additional malignancy that is progressing or has required active treatment within the past 3 years with the exception of basal cell carcinoma of the skin, squamous cell carcinoma of the skin, or carcinoma in situ (e.g. in situ cervical cancer or breast carcinoma) that have undergone potentially curative therapy
• Has radiographically detectable (even if asymptomatic and/or previously treated) central nervous system metastases and/or carcinomatous meningitis
• Has Grade ≥2 audiometric hearing loss
• Has Grade ≥2 neuropathy
• Has Grade 3-4 bleeding due to the underlying malignancy
• Has received major surgery or has not recovered adequately from the toxicity and/or complications from the intervention prior to study start
• Has had previous allogeneic tissue/solid organ transplant
• Has severe hypersensitivity (≥Grade 3) to pembrolizumab and/or any of its excipients, radiotherapy, cisplatin or their analogs
• Has an active autoimmune disease that has required systemic treatment in past 2 years
• Has a history of (non-infectious) pneumonitis that required steroids or has current pneumonitis
• Has an active infection requiring systemic therapy
• Has a known history of human immunodeficiency virus (HIV) infection
• Has a known history of or is positive for Hepatitis B (defined as hepatitis B surface antigen [HBsAg] reactive) or known active Hepatitis C (defined as Hepatitis C virus [HCV] ribonucleic acid is detected).
• Has a history or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the study, interfere with the participant's participation for the full duration of the study, or is not in the best interest of the participant to participate, in the opinion of the investigator
• Has a known psychiatric or substance abuse disorder that would interfere with the participant's ability to cooperate with the requirements of the study

Inclusion Criteria:

• STEP 1: Age >= 18 years
• STEP 1: Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1.
• STEP 1: Patients must have oropharynx cancer (American Joint Committee on Cancer [AJCC] 8) that is p16-positive by immunohistochemistry OR p16 equivocal by IHC and HPV positive by in situ hybridization with the following criteria: >= 10 pack-years, stage T1-2N2-N3 or T3-4N0-3 (less than 10 pack-years is considered a non-smoker) OR < 10 pack-years, stage T4N0-N3 or T1-3N2-3
• STEP 1: Patients must not have known hypersensitivity to nivolumab or compounds of similar chemical or biologic composition.
• STEP 1: Patients with a history of allergic reactions attributed to platinum-based chemotherapy agents are excluded.
• STEP 1: Patients must not have had prior systemic therapy, radiation treatment or surgery for p16 positive oropharyngeal squamous cell carcinoma (OPSCC).
• NOTE: Patients who had resection of T1 or T2 carcinoma with no radiation or chemotherapy are eligible if surgery was done 5 years prior to enrollment
• STEP 1: Patients must not have received previous irradiation for head and neck tumor, skull base, or brain tumors.
• STEP 1: Patients must not receive investigational agents within 4 weeks of enrollment or at any time while on study.
• STEP 1: Patients with evidence of distant metastases or leptomeningeal disease (LMD) are excluded.
• STEP 1: Patients with uncontrolled inter-current illnesses which in the opinion of the investigator will interfere with the ability to undergo therapy including chemotherapy are excluded.
• STEP 1: Patients with a history of prior or second malignancy are excluded, with the exception of curatively treated non-melanoma skin cancer, or curatively treated cervical cancer; additionally, patients curatively treated for malignancy who remain disease-free at > 2 years of follow up, are not excluded.
• STEP 1: Absolute neutrophil count (ANC) >= 1500/mm^3 (must be obtained =< 2 weeks prior to randomization).
• STEP 1: Hemoglobin (Hgb) >= 8.0 g/dL (must be obtained =< 2 weeks prior to randomization).
• STEP 1: Platelet count >= 100,000/mm^3 (must be obtained =< 2 weeks prior to randomization).
• STEP 1: Creatinine clearance of >= 60 ml/min (must be obtained =< 2 weeks prior to randomization). Creatinine clearance may be measured or calculated. If calculating, creatinine clearance, use the Cockcroft-Gault formula.
• STEP 1: Total bilirubin within 1.5 times the normal limits (must be obtained =< 2 weeks prior to randomization).
• STEP 1: Serum glutamic oxaloacetic transaminase (SGOT) (aspartate aminotransferase [AST]) or serum glutamate pyruvate transaminase (SGPT) (alanine aminotransferase [ALT]) within 2.0 times the normal limits (must be obtained =< 2 weeks prior to randomization).
• STEP 1: Alkaline phosphatase within 2.0 times the normal limits (must be obtained =< 2 weeks prior to randomization).
• STEP 1: Patients must not be pregnant or breast-feeding as chemotherapy, radiation, and immunotherapy may have possible teratogenicity effects; in addition, complications from pregnancy may interfere with the ability of patients to have an uninterrupted therapy. All patients of childbearing potential must have a blood test or urine study within 2 weeks prior to randomization to rule out pregnancy. A patient of childbearing potential is any female, regardless of sexual orientation or whether they have undergone tubal ligation, who meets the following criteria: 1) has achieved menarche at some point, 2) has not undergone a hysterectomy or bilateral oophorectomy or 3) has not been naturally postmenopausal (amenorrhea following cancer therapy does not rule out childbearing potential) for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months).
• STEP 1: Patients of childbearing potential must use an accepted and effective method of contraception or abstain from sexual intercourse for at least one week prior to the start of treatment, and continue for 5 months after the last dose of protocol treatment. Patients must also not donate ova during this same time period.
• STEP 1: Patients must have measurable disease
• STEP 1: Patients must have tumor measurements with CT of neck and CT of chest (or CT of neck and FDG PET/CT if standard of care) within 4 weeks prior to Step 1 randomization.
• STEP 1: Patients with active autoimmune disease or history of autoimmune disease that might recur, which may affect vital organ function or require immune suppressive treatment including systemic corticosteroids, should be excluded. These include but are not limited to patients with a history of immune related neurologic disease, multiple sclerosis, autoimmune (demyelinating) neuropathy, Guillain-Barre syndrome, myasthenia gravis; systemic autoimmune disease such as systemic lupus erythematosus (SLE), connective tissue disease, scleroderma, inflammatory bowel disease (IBD), Crohn's, ulcerative colitis, hepatitis; and patients with a history of toxic epidermal necrolysis (TEN), Stevens-Johnson syndrome, or phospholipid syndrome should be excluded because of the risk of recurrence or exacerbation of disease. Patients with vitiligo, endocrine deficiencies including thyroiditis managed with replacement hormones including physiologic corticosteroids are eligible. Patients with rheumatoid arthritis and other arthropathies, Sjogren's syndrome and psoriasis controlled with topical medication and patients with positive serology, such as antinuclear antibodies (ANA), anti-thyroid antibodies should be evaluated for the presence of target organ involvement and potential need for systemic treatment but should otherwise be eligible.
• STEP 1: Patients are permitted to enroll if they have vitiligo, type I diabetes mellitus, residual hypothyroidism due to autoimmune condition only requiring hormone replacement, psoriasis not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger (precipitating event).
• STEP 1: Patients must not have a condition requiring systemic treatment with either corticosteroids (> 10 mg/day prednisone equivalents) or other immunosuppressive medications which are expected to continue during nivolumab administration. Inhaled or topical steroids and adrenal replacement doses > 10 mg/day prednisone equivalents are permitted in the absence of active autoimmune disease
• STEP 1: Patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated
• STEP 1: Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load
• STEP 1: Patients with a known history of testing positive for human immunodeficiency virus (HIV) or known acquired immunodeficiency syndrome (AIDS) must have no detectable viral load on a stable antiviral regimen
• STEP 1: Patients must not be receiving any other investigational agents.
• STEP 1: Patient must not have a baseline clinically significant hearing loss, which in the opinion of the investigator would preclude the use of cisplatin
• STEP 2: Patients must have progression per RECIST criteria AND tissue-proven progression on Arm B treatment within 12 months after completion of radiation therapy.
• STEP 2: ECOG performance status of 0 or 1.
• STEP 2: Patients must not have known hypersensitivity to nivolumab or compounds of similar chemical or biologic composition.
• STEP 2: Patients must not have received non-protocol anti-cancer therapy after completion of radiation and chemotherapy.
• STEP 2: ANC >= 1500/mm^3 (must be obtained =< 2 weeks prior to registration).
• STEP 2: Hgb >= 8.0 g/dL (must be obtained =< 2 weeks prior to registration).
• STEP 2: Platelet count >= 100,000/mm^3 (must be obtained =< 2 weeks prior to registration).
• STEP 2: Creatinine within institutional limits of normal (must be obtained =< 2 weeks prior to registration)
• STEP 2: Total bilirubin within 1.5 times the normal limits (must be obtained =< 2 weeks prior to registration).
• STEP 2: SGOT (AST) or SGPT (ALT) within 2.0 times the normal limits (must be obtained =< 2 weeks prior to registration).
• STEP 2: Alkaline phosphatase within 1.5 times the normal limits (must be obtained =< 2 weeks prior to registration).
• STEP 2: Patients must not be pregnant or breast-feeding as chemotherapy, radiation, and immunotherapy may have possible teratogenicity effects; in addition, complications from pregnancy may interfere with the ability of patients to have an uninterrupted therapy. All women of childbearing potential must have a blood test or urine study within 2 weeks prior to registration to rule out pregnancy. A women of childbearing potential is any female, regardless of sexual orientation or whether they have undergone tubal ligation, who meets the following criteria: 1) has achieved menarche at some point, 2) has not undergone a hysterectomy or bilateral oophorectomy or 3) has not been naturally postmenopausal (amenorrhea following cancer therapy does not rule out childbearing potential) for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months).
• STEP 2: Patients of childbearing potential must use an accepted and effective method of contraception or abstain from sexual intercourse for at least one week prior to the start of treatment, and continue for 5 months after the last dose of protocol treatment. Patients must also not donate ova during this same time period.
• STEP 2: Patients must have measurable disease at the time of documented progression
• NOTE: For patients that have undergone salvage surgery for disease recurrence, measurable disease is not required at the time of registration to Step 2
• STEP 2: Patients must have tumor measurements with CT of neck and CT of chest (or CT of neck and FDG PET/CT if standard of care) within 4 weeks prior to Step 2 registration
• NOTE: Patients that have undergone salvage surgery for disease recurrence prior to Step 2 are not required to have measurable disease post-resection, but must have CT of neck and CT of chest (or CT of neck and FDG PET/CT if standard of care) after salvage surgery and within 4 weeks prior to step 2 registration to establish a baseline prior to nivolumab

Inclusion Criteria:

• Biopsy proven (from primary lesion and/or lymph nodes) diagnosis of cancer of the nasopharynx
• Sites are required to complete Step 1 registration before submitting specimens for EBV DNA analysis.
• Patients must have detectable pretreatment plasma EBV DNA, determined by the central lab prior to Step 2 registration (see Section 10.2 for details of specimen submission).
• For patients who have detectable plasma EBV DNA tested at one of the credentialed central labs (listed on the EBV DNA Testing Specimen Transmittal form) within 28 days prior to Step 1 registration: that test result can be used for eligibility without the need for re-testing. To use this test result for eligibility, the central lab must enter the test result through the pathology portal, and the site must follow the instructions in Section 5.4.
• Stage II-IVB disease (AJCC, 7th ed.) with no evidence of distant metastasis, based upon the following minimum diagnostic workup:
• History/physical examination by a Medical Oncologist or Clinical Oncologist or Radiation Oncologist or ENT, which must include an endoscopic evaluation, a complete list of current medications, and assessment of weight and weight loss in the past 6 months within 21 days prior to registration;
• Evaluation of tumor extent required within 28 days prior to registration:
• MRI of the nasopharynx and neck; or CT of the nasopharynx and neck with ≤ 3 mm contiguous slices with contrast and bone windows (to evaluate base of skull involvement). Note: If a treatment planning CT scan is used, it must be with ≤ 3 mm contiguous slices with contrast and be read by a radiologist. Please refer to section 6.3.2 for MRI requirement for target delineation.
• To rule out distant metastasis, patients must undergo the following imaging within 28 days prior to registration: 1. a CT scan with contrast of the chest and abdomen (required), and the pelvis (optional), or a total body PET/CT scan (non-contrast PET/CT is acceptable); 2. a bone scan only when there is suspicion of bone metastases (a PET/CT scan can substitute for the bone scan).
• Zubrod performance status 0-1 within 21 days prior to registration
• Absolute neutrophil count (ANC) ≥ 1,500 cells/mm^3
• Platelets ≥ 100,000 cells/mm^3
• Hemoglobin ≥ 8.0 g/dl (Note: the use of transfusion or other intervention to achieve hemoglobin [Hgb] ≥ 8.0 g/dl is acceptable)
• Total bilirubin ≤ 1.5 x institutional upper limit of normal (ULN)
• Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) ≤ 1.5 x institutional ULN
• Alkaline phosphatase ≤ 1.5 x institutional ULN
• Serum creatinine ≤ 1.5 mg/dl or calculated creatinine clearance (CC) ≥ 50 ml/min determined by 24-hour urine collection or estimated by Cockcroft-Gault formula
• Negative serum pregnancy test within 14 days prior to registration for women of childbearing potential
• Women of childbearing potential and male participants who are sexually active must agree to use a medically effective means of birth control throughout protocol treatment
• Patient must provide study specific informed consent prior to study entry, including the mandatory pre-treatment plasma EBV DNA assay

Exclusion Criteria:

• Prior invasive malignancy (except node negative, non-melanomatous skin cancer) unless disease free for a minimum of 1095 days (3 years) (for example, carcinoma in situ of the breast, oral cavity, or cervix are all permissible)
• Prior systemic chemotherapy for the study cancer; note that prior chemotherapy for a different cancer is allowable; however, at least 6-weeks recovery is necessary if the last regimen included nitrosourea or mitomycin
• Prior radiotherapy to the region of the study cancer that would result in overlap of radiation therapy fields
• Patients with hearing loss assessed to be primarily sensorineural in nature, requiring a hearing aid, or intervention (i.e. interfering in a clinically significant way with activities of daily living); a conductive hearing loss that is tumor-related is allowed
• ≥ Grade 2 peripheral sensory neuropathy (CTCAE, v. 4.0)
• Severe, active co-morbidity, defined as follows:
• Major medical or psychiatric illness, which in the investigator's opinion would interfere with the completion of therapy and follow up or with full understanding of the risks and potential complications of the therapy
• Unstable angina and/or uncontrolled congestive heart failure within the past 6 months
• Myocardial infarction within the last 6 months
• Acute bacterial or fungal infection requiring intravenous antibiotics at the time of registration; note that patients switched from IV antibiotics and currently on oral antibiotics whose infection is assessed to be adequately treated or controlled are eligible
• Chronic Obstructive Pulmonary Disease exacerbation or other respiratory illness requiring hospitalization or precluding study therapy within 30 days prior to registration
• Acquired Immune Deficiency Syndrome (AIDS) based upon current Centers for Disease Control and Prevention (CDC) definition; note, however, that Human Immunodeficiency Virus (HIV) testing is not required for entry into this protocol. The need to exclude patients with AIDS from this protocol is necessary because the treatments involved in this protocol may be significantly immunosuppressive.
• Pregnancy or women of childbearing potential and men who are sexually active and not willing/able to use medically acceptable forms of contraception
• Prior allergic reaction to the study drug(s) involved in this protocol
• Patients with undetectable pre-treatment plasma EBV DNA

Inclusion Criteria:

• Pathologically (histologically or cytologically) proven diagnosis of squamous cell carcinoma (including the histological variants papillary squamous cell carcinoma and basaloid squamous cell carcinoma but not neuroendocrine phenotype) of the oropharynx (tonsil, base of tongue, soft palate, or oropharyngeal walls); cytologic diagnosis from a cervical lymph node is sufficient in the presence of clinical evidence of a primary tumor in the oropharynx. Clinical evidence should be documented, may consist of palpation, imaging, or endoscopic evaluation, and should be sufficient to estimate the size of the primary (for T stage)
• Patients must have clinically or radiographically evident measurable disease at the primary site or at nodal stations. Simple tonsillectomy or local excision of the primary without removal of nodal disease is permitted, as is excision removing gross nodal disease but with intact primary site. Limited neck dissections retrieving =< 4 nodes are permitted and considered as non-therapeutic nodal excisions
• P16-positive based on local site immunohistochemical tissue staining (defined as greater than 70% strong diffuse nuclear or nuclear and cytoplasmic staining of tumor cells). Fine needle aspiration (FNA) biopsy specimens may be used as the sole diagnostic tissue. Centers are encouraged to contact the pathology chair for clarification
• Note: Institutions must screen patients, whose tumors must be p16-positive by immunohistochemistry (IHC) in order to be eligible for the trial using a Clinical Laboratory Improvement Amendments (CLIA)-certified laboratory. A rigorous laboratory accreditation process similar to the United States (U.S.) CLIA certification, such as the provincial accreditation status offered by the Ontario Laboratory Accreditation (OLA) Program in Canada, the College of American Pathologists (CAP), or an equivalent accreditation in other countries, is acceptable. The p16-positive results must be reported on the pathology report being submitted
• Clinical stage T1-2, N1, M0 (American Joint Committee on Cancer [AJCC], 8th edition [ed.]) or T3, N0-N1, M0 (AJCC, 8th ed.) including no distant metastases based on the following diagnostic workup:
• General history and physical examination within 56 days prior to registration;
• Exam with laryngopharyngoscopy (mirror or in office direct procedure acceptable) within 70 days prior to registration;
• One of the following imaging studies is required within 56 days prior to registration:
• FDG-PET/CT of the neck and chest (with or without contrast); FDG-PET/CT scan is strongly preferred and highly recommended to be used for eligibility OR
• Chest CT (with or without contrast)
• One of the following imaging studies is required within 28 days prior to registration:
• A diagnostic CT scan of neck (with contrast and of diagnostic quality) OR
• An magnetic resonance imaging (MRI) of the neck (with contrast and of diagnostic quality)
• Note: A diagnostic quality CT or MRI with contrast or FDG-PET/CT scan of neck performed for the purposes of radiation planning may serve as both staging and planning tools
• Patients must provide their personal smoking history prior to registration. The lifetime cumulative history cannot exceed 10 pack-years. The following formula is used to calculate the pack-years during the periods of smoking in the patient's life; the cumulative total of the number of pack-years during each period of active smoking is the lifetime cumulative history
• Number of pack-years = [Frequency of smoking (number of cigarettes per day) x duration of cigarette smoking (years)] / 20
• Note: Twenty cigarettes is considered equivalent to one pack. The effect of non-cigarette tobacco products on the survival of patients with p16-positive oropharyngeal cancers is undefined. While there are reportedly increased risks of head and neck cancer associated with sustained heavy cigar and pipe use (Wyss 2013), such sustained use of non-cigarette products is unusual and does not appear to convey added risk with synchronous cigarette smoking. Cigar and pipe tobacco consumption is therefore not included in calculating the lifetime pack-years. Marijuana consumption is likewise not considered in this calculation. There is no clear scientific evidence regarding the role of chewing tobacco-containing products in this disease, although this is possibly more concerning given the proximity of the oral cavity and oropharynx. In any case, investigators are discouraged from enrolling patients with a history of very sustained use (such as several years or more) of non-cigarette tobacco products alone
• Zubrod performance status of 0-1 within 14 days prior to registration
• Absolute neutrophil count >= 1,500/mcL (within 14 days prior to registration)
• Platelets >= 100,000/mcL (within 14 days prior to registration)
• Hemoglobin >= 8.0 g/dL (within 14 days prior to registration) (Note: use of transfusion or other intervention to achieve hemoglobin [Hgb] >= 8.0 g/dL is acceptable)
• Total bilirubin =< 1.5 x institutional upper limit of normal (ULN) (within 14 days prior to registration)
• Aspartate aminotransferase (AST)(serum glutamic-oxaloacetic transaminase [SGOT]) or alanine aminotransferase (ALT)(serum glutamate pyruvate transaminase [SGPT]) =< 3.0 x institutional ULN (within 14 days prior to registration)
• Serum creatinine =< 1.5 x ULN OR creatinine clearance (CrCl) >= 50 mL/min (if using the Cockcroft-Gault formula) (within 14 days prior to registration)
• Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial
• For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated
• Note: Known positive test for hepatitis B virus surface antigen (HBV sAg) indicating acute or chronic infection would make the patient ineligible unless the viral load becomes undetectable on suppressive therapy. Patients who are immune to hepatitis B (anti-hepatitis B surface antibody positive) are eligible (e.g. patients immunized against hepatitis B)
• Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment for the hepatitis, they are eligible if they have an undetectable HCV viral load.
• Note: Known positive test for hepatitis C virus ribonucleic acid (HCV RNA) indicating acute or chronic infection would make the patient ineligible unless the viral load becomes undetectable on suppressive therapy
• For women of childbearing potential (WOCBP), negative serum or urine pregnancy test within 24 hours prior to registration
• Women of childbearing potential (WOCBP) is defined as any female who has experienced menarche and who has not undergone surgical sterilization (hysterectomy or bilateral oophorectomy) or who is not postmenopausal. Menopause is defined clinically as 12 months of amenorrhea in a woman over 45 in the absence of other biological or physiological causes. In addition, women under the age of 55 must have a documented serum follicle stimulating hormone (FSH) level less than 40 mIU/mL
• Women of childbearing potential (WOCBP) and men who are sexually active with WOCBP must be willing to use an adequate method of contraception during and after treatment
• The patient or a legally authorized representative must provide study-specific informed consent prior to study entry
• Only English, Spanish, or French speaking patients are eligible to participate as these are the only languages for which the mandatory dysphagia-related patient reported instrument (MDADI) is available

Exclusion Criteria:

• Clinical stages T0; T4; T1-2, N0; or any N2 (AJCC, 8th ed)
• Recurrent disease
• Definitive clinical or radiologic evidence of metastatic disease or adenopathy below the clavicles
• Cancers considered to be from an oral cavity site (oral tongue, floor mouth, alveolar ridge, buccal or lip), or the nasopharynx, hypopharynx, or larynx, even if p16-positive, or histologies of adenosquamous, verrucous, or spindle cell carcinomas
• Carcinoma of the neck of unknown primary site origin (T0 is ineligible, even if p16-positive)
• Radiographically matted nodes, defined as 3 abutting nodes with loss of the intervening fat plane
• Supraclavicular nodes, defined as nodes centered below the level of the cricoid cartilage
• Gross total excision of both primary and nodal disease; this includes tonsillectomy, local excision of primary site, and nodal excision that removes all clinically and radiographically evident disease. In other words, to participate in this protocol, the patient must have clinically or radiographically evident gross disease for which disease response can be assessed
• Patients with simultaneous primary cancers or separate bilateral primary tumor sites are excluded with the exception of patients with bilateral tonsil cancers
• Prior invasive malignancy (except non-melanomatous skin cancer) unless disease free for a minimum of 1095 days (3 years) (of note, the exclusion applies only for invasive cancers such that carcinoma in situ of the breast, oral cavity, or cervix are all permissible)
• Prior systemic chemotherapy for the study cancer; note that prior chemotherapy for a different cancer is allowable
• Prior radiotherapy to the region of the study cancer that would result in overlap of radiation therapy fields
• Prior treatment with an anti-PD-1, anti-PD-L1, anti-PD-L2, anti-CTLA-4 antibody, or any other antibody or drug specifically targeting T-cell co-stimulation or immune checkpoint pathways
• History of severe hypersensitivity reaction to any monoclonal antibody.
• Severe, active co-morbidity defined as follows:
• Unstable angina and/or congestive heart failure requiring hospitalization within the last 6 months
• Transmural myocardial infarction within the last 6 months
• Acute bacterial or fungal infection requiring intravenous antibiotics at the time of registration
• Chronic obstructive pulmonary disease exacerbation or other respiratory illness requiring hospitalization or precluding study therapy within 30 days of registration
• Hepatic insufficiency resulting in clinical jaundice and/or coagulation defects
• Acquired immune deficiency syndrome (AIDS) based upon current Centers for Disease Control and Prevention (CDC) definition with immune compromise greater than that noted; note, however, that HIV testing is not required for entry into this protocol. The need to exclude patients with AIDS from this protocol is necessary because the treatments involved in this protocol may be significantly immunosuppressive. Protocol-specific requirements may also exclude immuno-compromised patients
• Condition requiring systemic treatment with either corticosteroids (> 10 mg daily prednisone equivalents) or other immunosuppressive medications within 14 days of registration. Inhaled or topical steroids and adrenal replacement doses < 10 mg daily prednisone equivalents are permitted in the absence of active autoimmune disease
• Patients with active autoimmune disease requiring systemic treatment (i.e. disease modifying agents, corticosteroids, or immunosuppressive drugs) should be excluded. These include but are not limited to patients with a history of immune related neurologic disease, multiple sclerosis, autoimmune (demyelinating) neuropathy, Guillain-Barre syndrome, myasthenia gravis; systemic autoimmune disease such as systemic lupus erythematosus (SLE), rheumatoid arthritis, connective tissue diseases, scleroderma, inflammatory bowel disease (IBD), Crohn's, ulcerative colitis, hepatitis; and patients with a history of toxic epidermal necrolysis (TEN), Stevens-Johnson syndrome, or phospholipid syndrome should be excluded because of the risk of recurrence or exacerbation of disease
• Note: Patients are permitted to enroll if they have vitiligo, type I diabetes mellitus, residual hypothyroidism due to autoimmune condition only requiring hormone replacement, psoriasis not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger (precipitating event)
• Patients who are pregnant, nursing, or expecting to conceive or father children
• Prior allergic reaction to cisplatin

Inclusion Criteria:

• PRIOR TO STEP 1 REGISTRATION INCLUSION:
• Pathologically (histologically or cytologically) proven diagnosis of squamous cell carcinoma of the oral cavity, including the oral (mobile) tongue, floor of mouth (FOM), mucosal lip, buccal mucosa, lower alveolar ridge, upper alveolar ridge, retromolar gingiva (retromolar trigone; RMT), or hard palate prior to registration
• Appropriate stage for study entry (T1-2N0M0; American Joint Committee on Cancer [AJCC] 8th edition [ed.]) based on the following diagnostic workup:
• History/physical examination within 42 days prior to registration
• Imaging of head and neck within 42 days prior to registration
• PET/CT scan or contrast neck CT scan, or gadolinium-enhanced neck magnetic resonance imaging (MRI) or lateral and central neck ultrasound; CT portion of the PET/CT must be of diagnostic quality
• Chest imaging with either a chest x-ray, CT chest scan (with or without contrast) or PET/CT (with or without contrast) within 42 days prior to registration
• Surgical assessment within 42 days prior to registration. Patient must be a candidate for surgical intervention with sentinel lymph node (SLN) biopsy and potential completion neck dissection (CND) or elective neck dissection (END)
• Surgical resection of the primary tumor will occur through a transoral approach with anticipation of resection free margins
• Zubrod performance status 0-2 within 42 days prior to registration
• For women of child-bearing potential, negative serum or urine pregnancy test within 42 days prior to registration
• The patient or a legally authorized representative must provide study-specific informed consent prior to study entry
• Only English-speaking patients (able to read and understand English) are eligible to participate as the mandatory patient reported NDII tool is only available in this language
• PRIOR TO STEP 2 RANDOMIZATION:
• FDG PET/CT required prior to step 2. Note: FDG PET/CT done prior to step 1 can be submitted for central review. However, if the FDG PET/CT is not of diagnostic quality, then FDG PET/CT will have to be repeated prior to Step 2 registration
• PET/CT node negative patients, determined by central read, will proceed to randomization.
• PET/CT positive patients will go off study, but will be entered in a registry and data will be collected to record the pathological outcome of neck nodes for diagnostic imaging assessment and future clinical trial development
• NOTE: All FDG PET/CT scans must be performed on an American College of Radiology (ACR) accredited scanner (or similar accrediting organization)

Exclusion Criteria:

• PRIOR TO STEP 1 REGISTRATION EXCLUSION:
• Definitive clinical or radiologic evidence of regional (cervical) and/or distant metastatic disease
• Prior non-head and neck invasive malignancy (except non-melanomatous skin cancer, including effectively treated basal cell or squamous cell skin cancer, or carcinoma in situ of the breast or cervix) unless disease free for ≥ 2 years
• Diagnosis of head and neck squamous cell carcinoma (SCC) in the oropharynx, nasopharynx, hypopharynx, and larynx
• Unable or unwilling to complete NDII (baseline only)
• Prior systemic chemotherapy for the study cancer; note that prior chemotherapy for a different cancer is allowable
• Prior radiotherapy to the region of the study cancer that would result in overlap of radiation therapy fields
• Patient with severe, active co-morbidity that would preclude an elective or completion neck dissection
• Pregnancy and breast-feeding mothers
• Incomplete resection of oral cavity lesion with a positive margin; however, an excisional biopsy is permitted
• Prior surgery involving the lateral neck, including neck dissection or gross injury to the neck that would preclude surgical dissection for this trial. Prior thyroid and central neck surgery is permissible; biopsy is permitted. Note: Borderline suspicious nodes that are ≥ 1 cm with radiographic finding suggestive of NOT malignant should be biopsied using ultrasound-guided (U/S-guided) fine-needle aspiration (FNA) biopsy
• Underlying or documented history of hematologic malignancy (e.g., chronic lymphocytic leukemia [CLL]) or other active disease capable of causing lymphadenopathy (sarcoidosis or untreated mycobacterial infection)
• Actively receiving systemic cytotoxic chemotherapy, immunosuppressive, anti-monocyte or immunomodulatory therapy
• Currently participating in another investigational therapeutic trial

Inclusion Criteria:

• Histologically proven squamous cell carcinoma of the oropharynx
• Clinical stage T1-2, N0-N3, M0 by American Joint Committee on Cancer (AJCC) 8 criteria
• Must have tumors deemed surgically resectable with acceptable morbidity
• Estimated life expectancy of at least 12 weeks
• Must give informed consent
• Must have Eastern Cooperative Oncology Group (ECOG) performance status =< 3
• Must have detectable circulating HPV DNA levels
• Platelets >= 100,000/ul
• Absolute neutrophil count (ANC) >= 1,500/ul
• Hemoglobin > 8 g/dl (use of transfusion to achieve this is acceptable)
• Total bilirubin < 2 X institutional upper limit of normal (ULN)
• Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase [SGOT])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase [SGPT]) < 3 X institutional ULN
• Serum creatinine < 2 x institutional ULN or creatinine clearance > 50 ml/min as determined by 24 hour collection or estimated by Cockcroft-Gault formula
• Negative pregnancy test, if applicable

Exclusion Criteria:

• Patients may not have received previous therapy for their head and neck squamous cell carcinoma (SCC), including chemotherapy, radiation therapy, or surgery beyond biopsy
• Second primary malignancy. Exceptions are:
• Patient had a second primary malignancy but has been treated and disease free for at least 3 years
• In situ carcinoma (e.g. in situ carcinoma of the cervix)
• Non-melanomatous carcinoma of the skin
• Patients with metastatic disease beyond the neck will be excluded
• Serious concomitant systemic disorders (including active infections) that would compromise the safety of the patient or compromise the patient's ability to complete the study, at the discretion of the investigator
• Age < 18 years
• Patients with human immunodeficiency virus (HIV) infection are not automatically excluded, but must meet the following criteria: CD4 count is > 499/cu mm and their viral load is < 50 copies/ml. Use of highly active antiretroviral therapy (HAART) is allowed
• Grade 3-4 electrolyte abnormalities (Common Terminology Criteria for Adverse Events [CTCAE], version [v.] 5)
• Serum calcium (ionized or adjusted for albumin) < 7 mg/dl (1.75 mmol/L) or > 12.5 mg/dl (> 3.1 mmol/L) despite intervention to normalize levels
• Magnesium < 0.9 mg/dl (< 0.4 mmol/L) or > 3 mg/dl (> 1.23 mmol/L) despite intervention to normalize levels
• Potassium < 3.0 mmol/L or > 6 mmol/L despite intervention to normalize levels
• Sodium < 130 mmol/L or > 155 mmol/L despite intervention to normalize levels
• Women who are pregnant, due to the teratogenic effects of radiation therapy and chemotherapy on the unborn fetus. Women of childbearing age must agree to undergo a pregnancy test prior to therapy and to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation and for 6 months after. Should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately
• Tumor deemed unresectable with acceptable morbidity:
• Tumors > 4 cm in size (T3 or higher)
• Tumors of the base of tongue < 4 cm but with deep invasion of tongue musculature placing hypoglossal nerve or both lingual arteries at risk
• Significant extension into hypopharynx
• Extension into soft palate beyond 1/3 of the width
• Clinically extensive extranodal extension (ENE) e.g. radiologic evidence of invasion of carotid artery, gross extension into sternocleidomastoid muscle or deep neck muscles. Lymph nodes larger than 6 cm without clinical ENE will be allowed