A Study of Avutometinib (VS-6766) and Defactinib in People With Mesonephric Gynecologic Cancer

Inclusion Criteria: * Female patients ≥ 18 years of age * Histologic confirmation of Gynecologic Mesonephric or Mesonephric-like cancer (GMC). Patients with mixed histology are eligible if the disease is deemed by the treating physician to be driven by the GMC component. * Measurable disease according to RECIST 1.1 * Patients must have persistent (disease that is metastatic at presentation or remains present following first-line therapy) or recurrent disease (disease that has come back or progressed following prior surgery or treatment) * Patients with metastatic or recurrent disease do not require any prior systemic therapy prior to enrollment. Patients may have received unlimited lines of prior systemic therapy. * Patients with treated brain metastases are eligible if follow-up brain imaging after CNS-directed therapy shows no evidence of progression. Patients with asymptomatic brain metastases that do not require intervention are also eligible. * HIV-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial. * 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. * 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. * Resolution of all toxicities of prior therapy or surgical procedures to baseline or Grade 1 (except for hypothyroidism requiring medication, and alopecia, which must have resolved to Grade ≤2). * Female patients with reproductive potential agree to use highly effective method of contraceptive during the trial and for 1 month following the last dose of study intervention. Hormonal forms of contraception are not recommended in this study. Non-hormonal methods of highly effective contraception include: * intrauterine device (IUD) * bilateral tubal occlusion * vasectomized partner * sexual abstinence * Patients must have adequate cardiac function with left ventricular ejection fraction ≥ 55% by echocardiography (ECHO) * Baseline QTc interval \< 460 ms (average of triplicate readings) (Common Terminology Criteria for Adverse Events \[CTCAE\] Grade 1) using Fredericia's QT correction formula. NOTE: This criterion does not apply to patients with a right or left bundle branch block. * Must have adequate organ function defined by the following laboratory parameters: * Total Bilirubin: ≤ 1.5 x institutional upper limit of normal (ULN) (patients with known Gilbert's disease who have bilirubin level ≤ 3 x ULN may be enrolled) * AST(SGOT)/ALT(SGPT): ≤3 × institutional ULN * Adequate renal function defined as either: Creatinine clearance (CrCL) or estimated Glomerular filtration rate (eGFR) of ≥50 mL/min estimated using either the Cockcroft-Gault equation, the Modification of Diet in Renal Disease Study, or as reported in the comprehensive metabolic panel/basic metabolic panel (eGFR). Or: Serum Creatinine ≤ 1.5 x ULN * Creatine phosphokinase (CPK) ≤ 2.5 x ULN. * Adequate hematologic function including: hemoglobin \[Hb\] ≥ 9.0 g/dL; platelets ≥ 100,000/mm\^3; and absolute neutrophil count \[ANC\] ≥ 1000/mm\^3 Exclusion Criteria: * Patients with newly diagnosed localized disease should be treated as per standard of care and are not eligible for this study. Patients who are candidates for potentially curative surgery or radiation are not eligible for this trial. * Systemic anti-cancer therapy (other than endocrine therapy) within 4 weeks, 1 cycle, or 5 half-lives (whichever is shortest) of the first dose of study intervention; Endocrine therapy within 1 week of the first dose of study intervention. * Major surgery within 4 weeks , minor surgery within 2 weeks, or palliative radiotherapy within 1 week of the first dose of study intervention. * Treatment with warfarin. Patients on warfarin for deep vein thrombosis/pulmonary embolism can be converted to low-molecular-weight heparin or direct oral anticoagulants (DOACs). * Prior treatment with a MEK or RAF or FAK inhibitor * Patients with the inability to swallow oral medications or impaired gastrointestinal absorption due to gastrectomy or drainage PEG tube * Patients with history of retinal pathology or evidence of visible retinal pathology that is considered a risk factor for RVO, intraocular pressure \> 21 mm Hg as measured by tonometry, or other significant ocular pathology, such as anatomical abnormalities that increase the risk for RVO * Patients with a history of corneal erosion (instability of corneal epithelium), corneal degeneration, active or recurrent keratitis, and other forms of serious ocular surface inflammatory conditions. * History of rhabdomyolysis * Patients with a history of hypersensitivity to any of the active or inactive avutometinib (VS-6766) and defactinib ingredients (hydroxypropylmethylcellulose, mannitol, magnesium stearate) of the investigational product. * Any other medical condition (e.g., cardiac, gastrointestinal, pulmonary, psychiatric, neurological, genetic, etc.) that in the opinion of the Investigator would places the patient at unacceptably high risk for toxicity. * Exposure to medications (with or without prescriptions), supplements, herbal remedies, or foods with potential for drug-drug interactions with study interventions within 14 days prior to the first dose of study intervention and during the course of therapy, including: * strong CYP3A4 inhibitors or inducers, due to potential drug-drug interactions with both avutometinib (VS-6766) and defactinib. * strong CYP2C9 inhibitors or inducers, due to potential drug-drug interactions with defactinib. * P-glycoprotein (P-gp) inhibitors or inducers, due to potential drug-drug interactions with defactinib. * strong breast cancer resistance protein (BCRP) inhibitors or inducers, due to potential drug-drug interactions with avutometinib and defactinib.