Pembrolizumab and Radiation for Head and Neck Cancer

Inclusion Criteria: * Pathologically confirmed either persistent and/ or locoregionally recurrent HNSCC of oral cavity, pharynx, larynx, unknown primary head and neck (H\&N) squamous cell carcinoma * Resectable disease as determined by the surgeon and team * Eastern Cooperative Oncology Group (ECOG) performance status (PS) \< 2 * At least 18 years of age * Adequate hematologic, renal, and hepatic function * Must have at least 2 week washout period from prior therapy * Willingness and ability to provide informed consent * Negative pregnancy test for females of reproductive potential * Patients who have undergone therapy for their cancer, such as surgery and/or chemotherapy and/or radiotherapy and recurred * Disease measurable by computed tomography (CT) or magnetic resonance imaging (MRI) per RECIST 1.1 criteria. * Prior definitive and palliative chemotherapy will be allowed * Prior radiation therapy will be allowed * Tumor tissue from resected site of disease must be provided for biomarker analyses, in addition to urine and blood sample as scheduled per protocol * White blood cell (WBC) \>= 2000/uL (obtained within 14 days of randomization) * Neutrophils \>= 1500/uL (obtained within 14 days of randomization) * Platelets \>= 100 x10\^3/uL (obtained within 14 days of randomization) * Hemoglobin \> 9.0 g/dL (obtained within 14 days of randomization) * Serum creatinine =\< 1.5 x upper limit of normal (ULN) or calculated creatinine clearance (CrCl) \>= 40 mL/min (Cockcroft and Gault or Wright formula may be used according to local practice) (obtained within 14 days of randomization) * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 3 x institutional ULN * Total Bilirubin =\< 1.5 x institutional ULN (except subjects with Gilbert Syndrome, who can have total bilirubin \< 3.0 mg/dL) * Women of childbearing potential (WOCBP) must use appropriate method(s) of contraception. WOCBP should use an adequate method to avoid pregnancy for four months after the last dose of pembrolizumab. * Women of childbearing potential must have a negative serum or urine pregnancy test (minimum sensitivity 25 IU/L or equivalent units of human chorionic gonadotropin \[HCG\]) * Men who are sexually active with WOCBP must use any contraceptive method with a failure rate of less than 1% per year. Men who are sexually active with WOCBP will be instructed to adhere to contraception for a period of four months after the last dose of investigational product Exclusion Criteria: * Requirement of immunosuppressive therapy within 14 days of randomization * Salivary gland carcinomas, lip carcinoma, adenocarcinoma of the skin * Prior use of immune checkpoint blockade agent * History of human immunodeficiency virus (HIV), hepatitis B, C: Participants who test positive for hepatitis B virus surface antigen (HBV sAg) or hepatitis C virus ribonucleic acid (HCV antibody) indicating acute or chronic infection, those who test positive for human immunodeficiency virus (HIV) or have known acquired immunodeficiency syndrome (AIDS) * Unresectable disease, as determined by the surgeon and team * Subjects with history of grade 3 toxicity with prior immunotherapy * Patients with distant metastases * Subjects with active autoimmune disease * Breastfeeding women * Additional prior malignancy within the previous 3 years (treated or untreated, except for skin carcinomas treated with excision alone and carcinoma in situ of the cervix) * Palliative radiotherapy less than 14 days prior to first dose of study drug * Any history of hypersensitivity to any of the trial medications * Poorly controlled or serious medical or psychiatric illness likely to interfere with participation and/or compliance in this clinical trial * Prisoners or subjects who are involuntarily incarcerated * Patients not available for follow-up/future contact as defined in the ICF * Note: Patients on this protocol are not excluded from participation in other clinical trials