Testing the Addition of M6620 (VX-970, Berzosertib) to Usual Chemotherapy and Radiation for Head and Neck Cancer

Inclusion Criteria: * Patients must have histologically or cytologically confirmed head and neck squamous cell cancer (HNSCC) including paranasal sinus cancers but excluding nasopharyngeal carcinomas * Clinical staged III or IV HNSCC, according to American Joint Committee on Cancer (AJCC) 7th Edition, that is not amenable to surgical resection * Carcinoma of the neck of unknown primary site origin (regardless of HPV/p16 status) is eligible * Age \>= 18 years; because no dosing or adverse event data are currently available on the use of M6620 (VX-970, berzosertib) in combination with cisplatin in patients \< 18 years of age, children are excluded from this study, but will be eligible for future pediatric trials * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Life expectancy of greater than 3 months * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm (\>= 2 cm) with conventional techniques or as \>= 10 mm (\>= 1 cm) with spiral computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 100,000/mcL * Total bilirubin within normal institutional limits * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanin…