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 combination antiretroviral therapy are eligible as long as they are on effective anti-retroviral therapy with undetectable viral load within 6 months and there is no drug-drug interaction with M6220 * For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, 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 * The effects of M6620 on the developing human fetus are unknown. For this reason and because DNA-damage response (DDR) inhibitors 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, for the duration of study participation, and for 6 months after completion of M6620 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 for 6 months after completion of M6620 administration * Patients with treated brain metastases are eligible if they are symptomatically stable while off steroid therapy for a minimum of 21 days * 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 * 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 * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Patients with symptomatic brain metastasis are not eligible due to their extremely poor prognosis and since it is unclear whether the investigational agent penetrates the blood-brain barrier. However, subjects who have had treatment for their brain metastasis and are symptomatically stable while off steroid therapy for a minimum of 21 days may be enrolled * Patients who have received prior topotecan therapy * Patients who have had chemotherapy or radiotherapy within 3 weeks prior to enrollment. * Note: Patients who have had palliative radiotherapy may be included as long as they have recovered from any radiotherapy related adverse events (allow at least a week between radiotherapy completion and study treatment) * Patients who have not recovered from adverse events due to prior anti-cancer therapy except hair loss and peripheral neuropathy (i.e., have residual toxicities \> grade 1) * Patients who are receiving any other investigational agents * History of allergic reactions attributed to compounds of similar chemical or biologic composition to M6620 or topotecan used in study * M6620 is primarily metabolized by cytochrome P450 3A4 (CYP3A4); therefore, concomitant administration with strong inhibitors of CYP3A4 (e.g., ketoconazole, itraconazole, clarithromycin, ritonavir, indinavir, nelfinavir and saquinavir) or inducers of CYP3A4 (e.g., rifampin, phenytoin, carbamazepine, phenobarbital, St. John's wort) should be avoided. Patients requiring any medications or substances that are strong inhibitors or inducers of CYP3A during the course of the study are ineligible. Because the lists of these agents are constantly changing, it is important to regularly consult a frequently-updated medical reference for a list of drugs to avoid or of which to minimize use. The Patient Drug Interactions Handout and Wallet Card should be provided to patients. As part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product * Patients with uncontrolled intercurrent illness * Pregnant women are excluded from this study because M6620 as a DDR inhibitor may have the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with M6620, breastfeeding should be discontinued if the mother is treated with M6620. These potential risks may also apply to topotecan used in this study * Patients with high grade neuroendocrine cancers, but with no small cell morphology will not be eligible * Patients with psychiatric illness/social situations that would limit compliance with study requirements * Patients with Li-Fraumeni syndrome will not be eligible