Combining immunotherapy and radiation for recurrent 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