Inclusion Criteria:
* Pathologically (histologically) proven diagnosis of HNSCC of the oral cavity, oropharynx, larynx, or hypopharynx prior to registration;
* Patients with oropharynx cancer need p16 determination by immunohistochemistry (where positive is defined as greater than 70% strong nuclear or nuclear and cytoplasmic staining of tumor cells), Note: Institutions must screen patients using a Clinical Laboratory Improvement Amendments (CLIA)-certified laboratory. A rigorous laboratory accreditation process similar to the United States (U.S.) CLIA certification, such as the provincial accreditation status offered by the Ontario Laboratory Accreditation (OLA) Program in Canada, the College of American Pathologists (CAP), or an equivalent accreditation in other countries, is acceptable. The p16 results must be reported on the pathology report being submitted;
* Oral cavity, larynx, hypopharynx, or p16-negative oropharynx cancer must be stages T1-2N2-3 or T3N1-3 or T4N0-3 (American Joint Committee on Cancer \[AJCC\] version 8);
* p16-positive oropharynx cancer patients, stages T4N0-3 or T1-3N2-3 (AJCC version 8);
* The patient has measurable disease as defined by the presence of at least one measurable lesion per RECIST 1.1;
* Please note: A histological or pathological specimen from cervical lymph nodes with well-defined primary site documented clinically or radiologically is acceptable
* Clinical stage noted above should be based upon following diagnostic workup:
* History/physical examination within 30 days prior to registration;
* Examination by radiation oncologist or medical oncologist or otolaryngology (ENT) or head \& neck surgeon 30 days prior to registration, including fiber optic exam with laryngopharyngoscopy;
* Diagnostic quality computed tomography (CT) or magnetic resonance imaging (MRI) of neck, with contrast, within 30 days prior to registration. Fludeoxyglucose F-18 (18F-FDG) whole body positron emission tomography (PET)-CT scan within 42 days of registration is strongly recommended but does not replace the CT or MRI study. Note: If CT component of the PET/CT is of diagnostic quality then PET/CT can be used for eligibility, however the PET/CT scan must be done within 30 days prior to registration
* Diagnostic quality, cross sectional imaging of the thorax within 42 days prior to registration; 18-F-FDG-PET/CT or conventional CT are acceptable
* Age \>= 18 years
* Patients must have a contraindication to cisplatin as defined in the following bullet points. Sites must complete the online tool at comogram.org prior to registration to determine if the patient is eligible. The scores must be recorded on a case report form (CRF). (Refer to data submission table on the NRG-HN008 protocol page on the NRG website);
* Age \>= 70 with moderate to severe comorbidity, defined as having one or more of the following conditions within 30 days prior to registration;
* Modified Charlson Comorbidity Index \>= 1
* Adult Comorbidity Evaluation (ACE)-27 Index \>= 1
* Omega score \< 0.80
* G-8 score =\< 14
* Cancer and Aging Research Group (CARG) Toxicity Score \>= 30%
* Cumulative Illness Rating Scale for Geriatrics (CIRS-G) Score \>= 4 OR
* Age \< 70 with severe comorbidity, defined as having two or more of the following conditions within 30 days prior to registration;
* Modified Charlson Comorbidity Index \>= 1
* ACE-27 Index \>= 1
* Omega score \< 0.80
* G-8 score =\< 14
* CARG Toxicity Score \>= 30%
* CIRS-G Score \>= 4 OR
* Age \>= 18 with an absolute or relative contraindication to cisplatin, defined as one or more of the following criterion within 30 days prior to registration:
* Pre-existing peripheral neuropathy grade \>= 1;
* Creatinine clearance (CrCl) must be \> 30 and \< 60 mL/min
* For this calculation, use the Cockcroft-Gault formula
* History of hearing loss, defined as either:
* Existing need of a hearing aid OR
* \>= 25 decibel shift over 2 contiguous frequencies on a pretreatment hearing test as clinically indicated
* Zubrod Eastern Cooperative Oncology Group (ECOG) performance status of 0-1 within 30 days prior to registration
* Whole blood cell (WBC) \>= 2000 cells/mm\^3 (within 30 days prior to registration)
* Absolute neutrophil count (ANC) \>= 1,500 cells/mm\^3 (within 30 days prior to registration)
* Platelets \>= 100,000 cells/mm\^3 (within 30 days prior to registration)
* Hemoglobin \>= 9.0 g/dL (within 30 days prior to registration); Note: The use of transfusion is acceptable
* Creatinine clearance (CrCl) \> 30 mL/min (within 30 days prior to registration)
* Total bilirubin =\< 1.5 x upper limit of normal (ULN) (except patients with Gilbert syndrome who can have total bilirubin \< 3.0 mg/dL) (within 30 days prior to registration)
* Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 2.5 x ULN (within 30 days prior to registration)
* For women of child bearing potential (e.g. uterus present and menstruating), a negative serum pregnancy test within 14 days prior to registration. Women of childbearing potential (WOCBP) is defined as any female who has experienced menarche and who has not undergone surgical sterilization (hysterectomy or bilateral oophorectomy) or who is not postmenopausal. Menopause is defined clinically as 12 months of amenorrhea in a woman over 45 in the absence of other biological or physiological causes. In addition, women under the age of 55 must have a documented serum follicle stimulating hormone (FSH) level less than 40 mIU/mL
* The patient must provide study-specific informed consent prior to study entry
* Known human immunodeficiency virus (HIV) infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months and CD4 T cell count \>= 200 are eligible for this trial. Testing is not required for entry into protocol
* Patients with a history of hepatitis B or C infection are eligible if they have an undetectable viral load
* Willing to use highly effective contraceptives for males and females of childbearing potential during therapy and for 12 weeks after the last dose of M3814 (peposertib); this inclusion is necessary because the treatment in this study may be significantly teratogenic
* Patients must be able to swallow whole tablets
Exclusion Criteria:
* Definitive clinical or radiologic evidence of distant (beyond cervical lymph node and neck tissue) metastatic disease
* Carcinoma of the neck of unknown primary site origin
* Patients with oral cavity cancer are excluded from participation if the patient is medically operable and the resection of the primary tumor is considered technically feasible by an oral or head and neck cancer surgical subspecialist
* Gross total excision of both primary and nodal disease; this includes tonsillectomy, local excision of primary site, and nodal excision that removes all clinically and radiographically evident disease
* Note: Patients with RECIST, version (v.) 1.1 evaluable remaining cancer either in the neck or primary site remain eligible
* Prior invasive malignancy (except non-melanomatous skin cancer carcinoma, in situ of the breast, oral cavity, or cervix, low or very low-risk prostate cancer) unless disease free for a minimum of 3 years
* Prior systemic chemotherapy for the study cancer; note that prior chemotherapy for a different cancer is allowable if not within =\< 3 years
* Prior radiotherapy to the region of the study cancer that would result in overlap of radiation therapy fields
* Severe, active co-morbidity defined as follows:
* History of bone marrow transplant and organ transplant, including allogenic stem cell transplantation;
* Unstable angina requiring hospitalization in the last 6 months;
* New York Heart Association Functional classification III/IV (Note: 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.);
* Myocardial infarction within the last 6 months;
* Persistent grade 3-4 (Common Terminology Criteria for Adverse Events \[CTCAE\] version 5.0) electrolyte abnormalities that cannot be reversed despite as indicated by repeat testing;
* Ongoing active infection that is associated with symptoms and/or requires antibiotic therapy at the time of registration (excluding asymptomatic bacteriuria, genital herpes, oral herpes, thrush, bacterial vaginosis, vaginal candidiasis, topical antifungals)
* Pregnancy and nursing females, if applicable
* Concomitant use of proton pump inhibitors (or unable to stop 5 days prior to treatment)
* Receipt of live vaccinations within 28 days prior to registration
* Patients unable to discontinue medications or substances that are:
* Strong inhibitors, inducers or sensitive substrates of CYP3A4/5, CYP2C19, or CYP2C9 prior to study treatment;
* Substrates of CYP1A2, CYP2B6, or CYP3A4/5 with a narrow therapeutic prior to study treatment;
* Note: Opioids are allowed, with the exception of methadone
* Fridericia's correction formula (QTcF) \> 450 ms for males and \> 470 ms for females
