Testing the Addition of an Anti-cancer Drug, BAY 1895344, With Radiation Therapy to the Usual Pembrolizumab Treatment for Recurrent Head and Neck Cancer

Inclusion Criteria: * Patients must have histologically confirmed recurrent or metachronous (second primary), unresectable head and neck squamous cell carcinoma, including oral cavity, oropharynx, larynx, hypopharynx, cutaneous, salivary gland, paranasal sinus, or cervical lymphadenopathy (head and neck cancer of unknown primary). Core needle biopsy (preferably at least three 18-gauge cores) or incisional biopsy is preferred over fine needle aspiration (FNA) for diagnosis of recurrent disease or new primary head and neck squamous cell carcinoma to provide sufficient tumor tissue for correlative studies. Unresectable refers both to patients who have declined surgery and patients deemed unresectable by otolaryngology. This includes patients for whom curative resection is medically contraindicated and/or would be associated with excessive surgical risk (as deemed by the consulting otolaryngologist) or undue surgical morbidity (e.g., total glossectomy, laryngectomy, and/or major resection requiring free flap reconstruction) * Patients must have either recurrent disease or a new primary squamous cell carcinoma of the head and neck within a previously irradiated area (radiotherapy to dose \>= 40 Gy, i.e., in-field recurrence) * Patients must have completed prior radiotherapy \>= 6 months prior to enrollment * Patients must meet at least one of the following criteria: 1) received prior platinum-containing chemotherapy; and/or 2) tumor expresses PD-L1 (Combined Positive Score \[CPS\…