• REGISTRATION ELIGIBILITY: Patients must have a history of pathologically confirmed locally advanced/unresectable Merkel cell carcinoma or metastatic Merkel cell carcinoma

• REGISTRATION ELIGIBILITY: Patients must have evaluable disease per Response Evaluation Criteria in Solid Tumors (RECIST) version (v)1.1

• REGISTRATION ELIGIBILITY: Patients must have had prior treatment with anti-PD-1 or anti-PD-L-1 antibody (e.g., pembrolizumab, avelumab, etc.) and have experienced progressive disease during treatment or within 120 days from the last dose of anti-PD-(L)1 therapy. Anti-PD-(L)1 therapy administered in combination with other agent(s) including ipilimumab is also allowed as prior therapy, if patients experienced progressive disease during treatment or within 120 days from the last dose of anti-PD-(L)1 therapy. If participants are receiving or received cytotoxic chemotherapy as most recent therapy prior to screening for this trial, there must be clinically and/or radiologically documented progressive disease on or after chemotherapy prior to being eligible for this study. If the patient is receiving bridging chemotherapy, the most recent administration must be ≥ 14 days prior to planned cycle 1 day 1 (C1D1) of the clinical trial to be eligible

• REGISTRATION ELIGIBILITY: Age \>= 18 years. Because no dosing or adverse event data are currently available on the use of M1774/tuvusertib in combination with avelumab in patients \< 18 years of age, children are excluded from this study

• REGISTRATION ELIGIBILITY: Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%)

• REGISTRATION ELIGIBILITY: Absolute neutrophil count \>= 1,000/mcL

• REGISTRATION ELIGIBILITY: Platelets \>= 100,000/mcL

• REGISTRATION ELIGIBILITY: Total bilirubin =\< institutional upper limit of normal (ULN) or ≤ 1.5 x ULN for subjects with Gilbert's disease

• REGISTRATION ELIGIBILITY: Aspartate aminotransferase (AST)(serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3 x institutional ULN

• REGISTRATION ELIGIBILITY: Creatinine =\< institutional ULN

• REGISTRATION ELIGIBILITY: Estimated glomerular filtration rate (eGFR) \>= 60 mL/min/1.73 m\^2

• REGISTRATION ELIGIBILITY: Hemoglobin \>= 9.0 g/dL

• REGISTRATION ELIGIBILITY: Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial

• REGISTRATION ELIGIBILITY: For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated

• REGISTRATION ELIGIBILITY: 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

• REGISTRATION ELIGIBILITY: Patients with treated brain metastases are eligible if follow-up brain imaging during screening shows no evidence of progressive brain metastases and it has been at least 4 weeks since central nervous system (CNS) directed therapy

• REGISTRATION ELIGIBILITY: 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

• REGISTRATION ELIGIBILITY: 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

• REGISTRATION ELIGIBILITY: The effects of M1774(tuvusertib) on the developing human fetus are unknown. For this reason and because ATR inhibitor agents as well as other therapeutic agents used in this trial are known to be teratogenic, women of child-bearing potential must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and, for the duration of study participation, and 6 months after completion of M1774 (tuvusertib) and avelumab administration. 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 3 months after completion of M1774 (tuvusertib) and avelumab administration

• REGISTRATION ELIGIBILITY: Ability to understand and the willingness to sign a written informed consent document. Legally authorized representatives may sign and give informed consent on behalf of study participants

• CROSSOVER ELIGIBILITY: Patient was initially assigned to arm 1 (M1774/tuvusertib monotherapy) and completed at least 21 of 28 possible doses of M1774/ tuvusertib

• CROSSOVER ELIGIBILITY: Patients must have documented progressive disease per RECIST v 1.1

• CROSSOVER ELIGIBILITY: ECOG performance status ≤ 2 (Karnofsky ≥ 60%)

• CROSSOVER ELIGIBILITY: Absolute neutrophil count ≥ 1,000/mcL (within 14 days of crossover registration)

• CROSSOVER ELIGIBILITY: Platelets ≥ 100,000/mcL (within 14 days of crossover registration)

• CROSSOVER ELIGIBILITY: Total bilirubin ≤ institutional upper limit of normal (ULN) or ≤ 1.5 x ULN for subjects with Gilbert's disease (within 14 days of crossover registration)

• CROSSOVER ELIGIBILITY: AST(SGOT)/ALT(SGPT) ≤ 3 × institutional ULN (within 14 days of crossover registration)

• CROSSOVER ELIGIBILITY: Creatinine ≤ institutional ULN (within 14 days of crossover registration)

• CROSSOVER ELIGIBILITY: Estimated glomerular filtration rate (eGFR) ≥ 60 mL/min/1.73 m\^2 (within 14 days of crossover registration)

• CROSSOVER ELIGIBILITY: Hemoglobin ≥ 9.0 g/dL (within 14 days of crossover registration)

• CROSSOVER ELIGIBILITY: Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial

• CROSSOVER ELIGIBILITY: For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated

• CROSSOVER ELIGIBILITY: 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

• CROSSOVER ELIGIBILITY: Patients with treated brain metastases are eligible if follow-up brain imaging during screening shows no evidence of progressive brain metastases and it has been at least 4 weeks since central nervous system (CNS) directed therapy

• CROSSOVER ELIGIBILITY: 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

• CROSSOVER ELIGIBILITY: 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

• CROSSOVER ELIGIBILITY: The effects of M1774 (tuvusertib) on the developing human fetus are unknown. For this reason and because ATR inhibitor agents as well as other therapeutic agents used in this trial are known to be teratogenic, women of child-bearing potential must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and, for the duration of study participation, and 6 months after completion of M1774 (tuvusertib) and avelumab administration. 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 3 months after completion of M1774 (tuvusertib) and avelumab administration