M6620 and Carboplatin With or Without Docetaxel in Treating Patients With Metastatic Castration-Resistant Prostate Cancer

Inclusion Criteria: * Participants must have histologically or cytologically confirmed prostate cancer (code 10036910) with progressive disease at the time of study entry by either * Sequence of at least 2 rising PSA values at a minimum of 1-week intervals * Radiographic progression per RECIST1.1 for soft tissue and/or per PCWG3\^2 for bone, with or without PSA progression * Patients must have metastatic disease by bone scan or other nodal or visceral lesions on computed tomography (CT) or magnetic resonance imaging (MRI) and a castrate level of testosterone (\< 50 ng/dL) and evaluable for disease response by either * Baseline PSA \>= 2.0 ng/mL OR * Measurable disease per RECIST 1.1 * NOTE: Subjects must maintain a castrate state; if they have not had an orchiectomy, they must continue to receive luteinizing hormone-releasing hormone (LHRH) or gonadotropin-releasing hormone (GnRH) agonists or antagonists unless intolerant * At least 2 prior treatments for castration resistant prostate cancer as follows: * Past progression or intolerance to at least one secondary hormonal therapy (abiraterone, enzalutamide, galeterone, apalutamide, darolutamide, orteronel, seviteronel or equivalent) * Past progression or intolerance to taxane-based chemotherapy * Eastern Cooperative Oncology Group (ECOG) performance status =\< 1 (Karnofsky \>= 70%) * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Hemoglobin \>= 9 g/dL (transfusion permitted) * Platelets \>= 150,000/mcL (without transfusion or growth factor in prior 28 days) * Total bilirubin =\< 1.5 x institutional upper limit of normal, unless the subject has known or suspected Gilbert's syndrome * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal or =\< 5 x if presence of liver metastases * Creatinine clearance \>= 40 mL/min/1.73 m\^2 * Prior treatment with mTOR inhibitors, cytostatic tyrosine kinase inhibitor (TKI), or biologic therapies allowed * Prior treatment with PARP inhibitors permitted * Patients with allergy or intolerance to docetaxel, grade 2 neuropathy are allowed on study, but if randomized to Arm A will receive carboplatin as a single agent (area under curve \[AUC\] 5) rather than docetaxel+carboplatin; they must be fit for carboplatin chemotherapy as determined by the treating investigator * Presence of a lesion (bone or soft tissue) amenable to image-guided percutaneous biopsy adequate for next generation sequencing (NGS), and planned to undergo core biopsy after trial registration but prior to cycle 1 day 1 of therapy; confirmation of adequacy of this biopsy material for NGS is NOT required for initiation of therapy; if elective biopsies are not being performed at the treating institution due to preparations or precautions related to coronavirus disease 2019 (COVID-19), this requirement can be waived on discussion with the trial principal investigator (PI) * The effects of M6620 (VX-970, berzosertib), carboplatin and docetaxel on the developing human fetus are unknown. For this reason and because DNA-damage response inhibitors and chemotherapeutic agents are known to be teratogenic, men 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. Should a woman become pregnant or suspect she is pregnant while 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 6 months after completion of carboplatin and M6620 (VX-970, berzosertib) administration * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to planned cycle 1 day 1 of study treatment; patients on an oral anti-neoplastic such as an oral hormonal agent, PARP inhibitor or oral experimental agent should discontinue \>= 14 days prior to planned cycle 1 day 1 of study treatment * Patients who have not recovered from adverse events due to prior anti-cancer therapy (i.e., have residual toxicities \> grade 1), except for grade 2 anorexia, alopecia, neuropathy, and fatigue, for which resolution is not required * Patients who are receiving any other investigational agents * Patients with known brain metastases or leptomeningeal disease should be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events * History of allergic reactions attributed to compounds of similar chemical or biologic composition to M6620 (VX-970, berzosertib) or carboplatin; (patients with allergy to docetaxel will be allowed on study, but docetaxel will be excluded from their treatment regimen) * Subjects receiving treatment with ototoxic or nephrotoxic medications that cannot be discontinued at least 7 days before first dose of carboplatin and for the duration of the study; inadvertent or short-term use on study will not cause a subject to be ineligible; if a short course of therapy with nephrotoxic or ototoxic medication is anticipated and required, carboplatin should be discontinued until 7 days after this course is completed; patients on continuous medications that are potentially nephrotoxic who have had no evidence of nephrotoxicity from these medications at study entry are allowed to continue those medicines on trial * M6620 (VX-970, berzosertib) is primarily metabolized by 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) is prohibited; 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 minimize use of; Patient Drug Information 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 * 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 and nursing women are excluded from this study because they do not develop prostate cancer * Human immunodeficiency (HIV)-positive participants with detectable viral load and/or CD4 count =\< 300 are ineligible due to increased risk of lethal infections when treated with marrow-suppressive therapy; HIV-positive patients with undetectable viral loads and CD4 counts \> 300, and not on interacting antiretroviral therapy may be eligible after discussing with the principal investigator * Prior treatment with platinum-containing regimen or ATR inhibitor for prostate cancer