Talazoparib and Axitinib for People With Previously Treated Advanced Kidney Cancer

Inclusion Criteria: * Biopsy proven metastatic or unresectable renal cell carcinoma with clear cell component * Prior treatment with at least 1 VEGFR TKI and 1 PD-1/PD-L1 immune checkpoint inhibitor (ICI).Combination VEGFR TKI plus ICI will be counted as 1 line of therapy. During the dose escalation portion of the study prior TKI exposure is not required. * Dose escalation portion: No maximum prior lines of therapy. Dose expansion portion: maximum of two prior lines of therapy * Adequate Hematologic Function * Absolute Neutrophil Count ≥ 1.5 x 109 / L * Platelet Count ≥ 100 x 10\^9 / L * Hemoglobin ≥ 9 g/dL * No transfusion of packed red blood cells or platelets within 21 days of Cycle 1 Day 1 * Adequate Renal Function ≥ 60 ml/min according to the Cockcroft-Gault formula ° Patients with moderate renal impairment (creatinine clearance 30-59 ml/min by Cockcroft-Gault) may be eligible in the phase II dose expansion * Adequate Hepatic Function including: * Total bilirubin ≤ 1.5 x upper limit of normal (ULN) * AST ≤ 3 x upper limit of normal (ULN) without liver metastasis * ALT ≤ 3 x upper limit of normal (ULN) without liver metastasis * AST or ALT ≤ 5 x upper limit of normal (ULN) for patients with liver metastasis * Patients with known Gilbert's syndrome may be included if total bilirubin ≤ x 3 ULN * Eastern Cooperative Group (ECOG) Performance Status 0-2. * Patients must have measurable disease by RECIST v1.1. At least one measurable lesion should not have been previously irradiated. * Women of childbearing potential must have negative urine or serum pregnancy testing at screening. All women will be considered childbearing potential unless meeting criteria including: * Achieved post-menopausal status as defined by cessation of regular menses for at least 12 consecutive months with no alternative pathological or physiological cause and have follicular stimulation hormone showing postmenopausal state. Women who have been amenorrhoeic for ≥ 12 months are still considered to be of childbearing potential if the amenorrhea is possibly due to prior chemotherapy, anorexia, low body weight, ovarian suppression, anti-estrogen therapy or other medically inducible reasons. * Documented hysterectomy or bilateral oophorectomy surgery * Medically confirmed ovarian failure * Sexually active participants and their partners must agree to use medically accepted methods of contraception (i.e. barrier methods including condoms, female condom, or diaphragm with spermicidal gel) during the study and for 7 months after the last dose of the study treatment for women, and 4 months for men. * Recovery of baseline CTCAE v5.0 grade ≤1 toxicities related to prior study treatments unless adverse events are clinically non-significant per investigator's discretion and/or stable on supportive therapy if needed. * Patients must be willing and able to comply with trial protocol. This includes adhering to the treatment plan, scheduled visits, laboratory and other study procedures. Participant Inclusion Criteria for Phase II Dose Expansion * Maximum 2 lines of prior therapy. Combination VEGFR TKI plus ICI therapy will be accepted as 1 line of therapy. * Adequate Renal Function ≥ 30 ml/min according to the Cockcroft-Gault formula. * Patients with normal renal function (Cr clearance ≥ 60 ml/min by Cockcroft-Gault formula) will receive talazoparib at RP2D. * Patients with moderate renal impairment (Cr clearance 30-59 ml/min by Cockcroft-Gault formula) will receive talazoparib at one dose level lower than RP2D. Exclusion Criteria: * Prior treatment with talazoparib or other agents which target PARP * Prior treatment with axitinib. Other VEGFR TKIs are permissible. * Patients \< 18 years old * Patients who are pregnant or breast-feeding. Fertile patients who are unwilling or unable to use two methods of contraception (at least one of which considered highly effective) for duration of study and after study (7 months after last dose of the study treatment for women, and 4 months for men) * Prior diagnosis of myelodysplastic syndrome (MDS) or diagnosis of other malignancy that requires anti-cancer directed therapy within the last 12 months. Exclusions include those cancers that are considered cured by local therapy (e.g. basal cell carcinoma, squamous cell carcinoma, ducal carcinoma in situ of breast, bladder of cervix) or other cancers that have low malignant potential and do not require systemic therapy (i.e. Gleason-grade \<6 prostate adenocarcinoma, borderline ovarian malignancy / low malignant potential). * Treatment with anti-cancer therapies within 21 days or five half-lives, whichever shorter, of start date, including monoclonal antibody, cytotoxic therapy, or another investigational agent. * Significant vascular disease (i.e. aortic aneurysm requiring surgical repair, recent arterial thrombosis) within 6 months prior to first dose of therapy. * Ejection Fraction (EF) ≤50% by echocardiogram (ECHO). Multi-gated acquisition scan (MUGA) should be obtained to estimate EF if quality of ECHO is insufficient. * Uncontrolled hypertension defined as systolic blood pressure (BP) ≥160 mmHg or diastolic BP ≥ 100 mmHg despite anti-HTN therapy. * Evidence of bleeding diathesis or significant unexplained coagulopathy (i.e. absent of anticoagulation) * Clinical signs or symptoms of gastrointestinal obstruction requirement parenteral hydration, parenteral nutrition, or feeding tube. * Uncontrolled effusion management (pleural effusion, pericardial effusion, or ascites) which requires recurrent drainage procedures. * Active infection requiring parenteral antibiotic therapy. * History of either positive HCV RNA viral load or anti-HCV antibody screening detectable; HBV infection with HBV surface antigen detection and/or positive HBV DNA viral load. * Known hypersensitivity to talazoparib or axitinib, or any component in formulations. * Severe acute or chronic medical conditions which may significantly increase the risk of study participants, per treating investigator's discretion. * Radiation therapy to any site (including bone) \<2 weeks prior to the first dose of therapy * Symptomatic brain metastasis or leptomeningeal disease requiring steroid use. Patients are eligible if they are neurologically stable for 4 weeks, have completed radiation therapy or surgery, and recovered from side effects. Patients must have discontinued steroid therapy for at least 2 weeks prior to first dose of study treatment. * Current or anticipated use of potent P-gp inhibitors within 7 days prior to randomization or anticipated use during the study * Inability to swallow capsules, known intolerance to talazoparib and axitinib or its excipients, known malabsorption syndrome, or other conditions which impair intestinal absorption.