Testing What Happens When an Immunotherapy Drug (Pembrolizumab) is Given by Itself Compared to the Usual Treatment of Chemotherapy With Radiation After Surgery for Recurrent Head and Neck Squamous Cell Carcinoma

Inclusion Criteria: * Patient must be between 18 and 79 years of age * Patient must have locoregionally recurrent or second primary HNSCC (oral cavity, oropharynx, larynx, hypopharynx) in a previously radiated field * Patient must have undergone surgery with gross total resection and must be randomized within 8 weeks of surgery * Patients must have high risk disease defined as: * Positive margins and/or extra nodal extension (ENE) * Positive margins are defined as malignancy at or within 1 mm of the margin. High grade dysplasia (i.e. carcinoma in situ) at the margin is also considered positive * ENE may be either gross or microscopic * Patient must have a PD-L1 Combined Positive Score (CPS) \>= 1 in a Clinical Laboratory Improvement Act (CLIA) certified laboratory. Testing can be done locally as long as it is done in a CLIA certified laboratory. This testing must be on the tumor specimen from the resection of the patient's recurrent or second primary HNSCC * Patient must have had prior radiation to the area of recurrent or second primary tumor. This is defined as \> 50% of the presurgical tumor volume having previously received a dose of \> 45 Gy as determined by the treating radiation oncologist * Patient must have completed prior radiation a minimum of 6 months prior to randomization * Patient must not have any evidence of distant disease based on baseline imaging done within 28 days prior to randomization * Patient must not have received anti-PD-1/PD-L1 therap…