A Phase I/II Study of Sacituzumab Govitecan Plus Berzosertib in Small Cell Lung Cancer, Extra-Pulmonary Small Cell Neuroendocrine Cancer and Homologous Recombination-Deficient Cancers Resistant to PARP Inhibitors

* INCLUSION CRITERIA: All phases and cohorts -Subjects must not have received chemotherapy or undergone major surgery within 2 weeks and radiotherapy within 24 hours prior to cycle 1 day 1. * Age \>=18 years. Because no dosing or adverse event data are currently available on the use of sacituzumab govitecan in combination with and berzosertib in participants \<18 years of age, children are excluded from this study, but will be eligible for future pediatric trials. * ECOG performance status \<=2 * Participants must have adequate organ and marrow function as defined below: * leukocytes \>=3,000/mcL * Hemoglobin \>=9.0g/dL * absolute neutrophil count \>=1,500/mcL * platelets \>=100,000/mcL * total bilirubin within normal institutional limits * AST(SGOT)/ALT(SGPT) \<=2.5 X institutional upper limit of normal * creatinine within normal institutional limits OR --creatinine clearance \>=30 mL/min/1.73 m2 for participants with creatinine levels above institutional normal (calculated by Cockcroft-Gault formula). * Participants with neurologically stable brain metastases defined as asymptomatic metastasis, or treated metastasis having no evidence of progression or hemorrhage for at least 2 weeks after treatment (including brain radiotherapy) may be included. Participants must be off any systemic corticosteroids for the treatment of brain metastases for at least 7 days prior to enrollment. * Participants with previously treated with topoisomerase 1/2 inhibitors can be enrolled. * The effects of the combined study drugs on the developing human fetus are unknown. For this reason and because study agents as well as other therapeutic agents used in this trial are known to be teratogenic, individuals of child-bearing potential and individuals who can father children must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, while taking the study drugs and for 6 months after the administered study drug (berzosertib or sacituzumab govitecan). * Ability of subject to understand and the willingness to sign a written informed consent document. Phase I -Participants with histologically or cytologically confirmed advanced solid tumors with progression on at least one prior chemotherapy. Phase II HRD cohort -Known HRD cancer and documented evidence of at least ONE or MORE of the following: --Pathogenic or likely pathogenic somatic mutation or inactivating alteration of a gene involved in homologous recombination (BRCA1, BRCA2, ATM, BRIP1, BARD1, CDK12, CHEK1, CHEK2, FANCL, PALB2, PPP2R2A, RAD51B, RAD51C, RAD51D, or RAD54L) repair in the tumor. Local testing in Clinical Laboratory Improvement Act (CLIA)-certified laboratory will be accepted. No variants of uncertain significance (VUS) will be allowed. * If this alteration is identified in a circulating tumor deoxyribonucleic acid (ctDNA) assay, the variant-allele fraction must be \> 20% to indicate relevance to predominant tumor clone * Mutation in one or more other genes involved in homologous DNA recombination repair in the tumor may be included at investigator's discretion ---Homologous recombination repair deficiency by genomic signature in the tumor by BROCA-HR, Foundation One or equivalent assay --Presence of pathogenic or likely pathogenic germline mutation/variant in BRCA1, BRCA2, ATM, BRIP1, BARD1, CDK12, CHEK1, CHEK2, FANCL, PALB2, PPP2R2A, RAD51B, RAD51C, RAD51D, or RAD54L. Germline mutations in other HR genes will be considered at investigator's discretion. Local testing in Clinical Laboratory Improvement Act (CLIA)-certified laboratory will be accepted. No variants of uncertain significance (VUS) will be allowed. * Participants must have measurable disease, per RECIST 1.1. * Participants must have at least one lesion deemed safe to biopsy and be willing to undergo mandatory biopsies (Pre-treatment, On-Treatment, PD). Up to 10 participants with non-biopsiable disease may be enrolled. * Participants should have demonstrated progressive disease while taking a PARPi as a previous therapy or within 6 months of completing PARPi therapy. Response to prior PARPi is not required. * Participants may have received chemotherapy in the interval between PARPi and enrollment. Phase II SCLC and EP-SCNC * Recurrent histologically or cytologically confirmed SCLC or EP-SCNC after at least one prior platinum-based therapy. * Participants must have measurable disease, per RECIST 1.1. * Participants must have at least one lesion deemed safe to biopsy and be willing to undergo mandatory biopsies (Pre-Treatment, On-Treatment, PD). Up to 10 participants with non-biopsiable disease may be enrolled. EXCLUSION CRITERIA: * Participants who are receiving any other investigational agents or concurrent systemic anti-cancer therapies. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to study drugs or other agents used in study. * Participants with symptomatic brain metastasis. * Participants who require treatment with strong inhibitors or inducers of CYP3A or with UGT1A1 inhibitors during the planned period of investigational treatment with sacituzumab govitecan. * Participants known to be homozygous for the UGT1A1\*28 variant allele with severely reduced UGT1A1 activity. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. * Pregnant individuals are excluded from this study because study drugs 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 study drugs, breastfeeding should be discontinued if the mother is treated with the study drugs. These potential risks may also apply to other agents used in this study. * Participants with myelosuppressive disorders or acute myeloid leukemia. * HIV positive participants with the following exception: Patients with long-standing (\>5 years) HIV on antiretroviral therapy \> 1 month (undetectable HIV viral load and CD4 count \> 150 cells/microL) may be eligible if the principal investigator determines no anticipated clinically significant drug-drug interactions. * Participants with evidence of chronic hepatitis B virus (HBV) infection, unless the HBV viral load is undetectable and participant is on suppressive therapy, if indicated. * HCV infected participants with the following exceptions: Participants with a history of hepatitis C virus (HCV) infection must have been treated and cured. For participants with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load.