• 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 \[CPS\] \>= 1)

• Patients who have received anti-PD-1, anti-PD-L1, or anti-PD-L2 therapy must have had disease progression while on this therapy. Patients who have not received prior anti-PD-1, anti-PD-L1, or anti-PD-L2 therapy are also eligible

• Age \>= 18 years. Because no dosing or adverse event data are currently available on the use of BAY 1895344 with pembrolizumab in patients \< 18 years of age, children are excluded from this study

• Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 50%) and a life expectancy of \>= 3 months

• Leukocytes \>= 3,000/mcL

• Absolute neutrophil count \>= 1,500/mcL

• Platelets \>= 100,000/mcL

• Hemoglobin \>= 9 g/dL or 5.6 mmol/L without transfusion or erythropoietin dependency (within 7 days of assessment)

• Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN) (If total bilirubin \> 1.5 x ULN, direct bilirubin must be \< ULN)

• Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional ULN (=\< 5 x ULN for patients with liver metastases)

• Creatinine OR measured or calculated creatinine clearance (CrCl) \< 1.5 x institutional ULN OR glomerular filtration rate (GFR) \> 60 mL/min/1.73 m\^2

‣ CrCl should be calculated per institutional standard

• Albumin \> 2.5 mg/dL

• International normalized ratio (INR) or prothrombin time (PT) =\< 1.5 x ULN unless patient is receiving anticoagulant therapy as long as PT or partial thromboplastin time (PTT) is within therapeutic range of intended use of anticoagulants

• Activated partial thromboplastin time (aPTT) =\< 1.5 x ULN unless patient is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants

• Patients must have measurable disease (at least one measurable lesion) as per Response Evaluation Criteria in Solid Tumors (RECIST) 1.1. Baseline imaging must include neck CT (preferably contrast-enhanced) and chest CT or skullbase to midthigh PET/CT (preferably with contrast-enhanced neck CT if diagnostic contrast-enhanced neck CT not available). Patients who have undergone surgery aside from biopsy may be included if gross disease is present within the surgical resection bed or at another site

• Patients must have no contraindications to pembrolizumab, including no history of organ allograft transplantation or active autoimmune disease (active defined as having autoimmune disease-related symptoms and detectable autoantibodies) that has required systemic treatment in the past 2 years (i.e., use of disease-modifying agents, corticosteroids, or immunosuppressive drugs). Replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment

• Human immunodeficiency virus (HIV)-infected patients may participate IF they meet the following eligibility requirements:

‣ They must be stable on their anti-retroviral regimen, and they must be healthy from an HIV perspective

⁃ They must have a CD4 count of greater than 250 cells/mcL

⁃ They must not be receiving prophylactic therapy for an opportunistic infection

• For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy (e.g., not hepatitis B surface antigen \[HBsAg\] reactive), if indicated

• 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 (e.g., HCV ribonucleic acid \[RNA\] \[qualitative\] is not detected)

• Patients with previously treated brain metastases may participate provided they are stable (without evidence of progression by imaging using the identical imaging modality for each assessment, either magnetic resonance imaging \[MRI\] or computed tomography \[CT\] scan, for at least 4 weeks prior to the first dose of trial treatment and any neurologic symptoms have returned to baseline), have no evidence of new or enlarging brain metastases, and are not using steroids for at least 7 days prior to trial treatment

• Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial (e.g. basal cell carcinoma, early-stage differentiated thyroid carcinoma, low-risk prostate cancer, ductal carcinoma in situ of the breast, squamous cell carcinoma of the skin that has undergone potentially curative therapy, in situ cervical cancer). Patients with 10 or fewer distant metastases may be eligible for this trial if they meet performance status inclusion criteria

• Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association functional classification. To be eligible for this trial, patients should be class 2B or better

• Enrolling site must perform PD-L1 testing on tumor biopsy per institutional protocol (local testing or tissue sent to outside laboratory for PD-L1 immunohistochemistry (IHC) 22C3 pharmDx). Combined positive score (CPS) is recommended for PD-L1 scoring. Test result does not need to be available at the time of enrollment and treatment initiation, unless the patient has not received prior platinum-containing chemotherapy. For patients without a history of platinum-containing chemotherapy, PD-L1 testing with CPS \>= 1 is required to be eligible to receive pembrolizumab

• Female patients of childbearing potential must have a negative urine or serum pregnancy test within 72 hours prior to receiving the first dose of study medication. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required. The effects of BAY 1895344 on the developing human fetus are unknown. For this reason and because DNA-damage response inhibitors as well as other therapeutic agents used in this trial are known to be teratogenic, women of child-bearing potential and men must agree to use adequate contraception prior to study entry, for the duration of study participation, and for 6 months after completion of BAY 1895344 therapy or 4 months after completion of pembrolizumab therapy, whichever is later. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 6 months after completion of BAY 1895344 therapy or 4 months after completion of pembrolizumab therapy, whichever is later

‣ Note: Abstinence is acceptable if this is the usual lifestyle and preferred contraception for the patient

• Ability to understand and the willingness to sign a written informed consent document. Participants with impaired decision-making capacity (IDMC) who have a legally-authorized representative (LAR) and/or family member available will also be eligible