Randomized Trial of Topotecan With M6620, an ATR Kinase Inhibitor, in Small Cell Lung Cancers and Small Cell Cancers Outside of the Lungs

Inclusion Criteria: * Patients enrolled to the primary cohort must have limited- or extensive-disease SCLC at diagnosis, with relapse at study entry with measurable disease at random assignment per Response Evaluation Criteria in Solid Tumors (RECIST) 1.1. Both platinum-sensitive and platinum-resistant patients will be included * Patients with extrapulmonary small cell cancers (cancers with small cell morphology and arising outside the lung, such as small cell prostate, bladder, etc.) will be eligible for the exploratory cohort * Patients must be \>= 18 years of age because no dosing or adverse event data are currently available on the use of M6620 in combination with topotecan in patients \<18 years of age * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Hemoglobin \>= 9.0 g/dL - patients may receive transfusion to meet the hemoglobin (Hb) eligibility * Absolute neutrophil count (ANC) \>= 1,500/mcL * Platelets \>= 100,000/mcL * Total bilirubin =\< 2 mg/dL * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3.0 x institutional upper limit of normal (ULN) * Creatinine =\< institutional ULN OR * Glomerular filtration rate (GFR) \>= 60 mL/min/1.73 m\^2 unless data exists supporting safe use at lower kidney function values, no lower than 30 mL/min/1.73 m\^2 * Human immunodeficiency virus (HIV)-positive subjects on combina…