A Study of Tinengotinib (TT-00420) in Combination With Standard Treatments in People With Prostate Cancer

Inclusion Criteria: * Participants ≥ 18 years old, with signed informed consent * Histologically confirmed carcinoma of the prostate (neuroendocrine differentiation is allowed, but pure small cell carcinoma is not permitted) * Metastatic disease documented by at least 2 bone lesions on whole body radionuclide bone scan, or soft tissue disease documented by computed tomography (CT) scan/magnetic resonance imaging (MRI). Note: Metastatic disease seen only on PET imaging does not qualify. * Current ongoing therapy and observed tolerance with full standard dose of abiraterone acetate (1000 mg QD) or enzalutamide (160 mg QD) at the time of study entry. Enzalutamide or abiraterone acetate must have been started at least 90 days before screening assessments. An interruption of dosing of a maximum of 30 days is permitted prior to resuming the agent. Please note: Patients who are on a reduced dose or are intolerant of abiraterone acetate or enzalutamide at screening will not be eligible for study participation. * Progressive disease on enzalutamide or abiraterone acetate documented by PCWG3 criteria for study entry. Progressive disease is defined as at least one of the following: 1. PSA progression defined as a minimum of 2 rising PSA levels with a minimum of a 1-week interval between each determination, reaching a minimum PSA value of 1.0 ng/mL. 2. Nodal or visceral progression as defined by PCWG3-modified RECIST 1.1 3. Appearance of 2 or more new lesions on a bone scan * At least one of the following at study entry: 1. RECIST 1.1 measurable disease at baseline; i.e., soft tissue tumor lesions or pathologically enlarged lymph nodes that can be accurately measured in at least one dimension OR 2. a PSA of 2.0 ng/mL or above * Participants must be medically or surgically castrated with ongoing androgen deprivation therapy (ADT) for ≥90 days or have documented history of bilateral orchiectomy. * ECOG 0 - 2 * Adequate organ function confirmed at screening, as evidenced by: * Absolute neutrophil count ≥ 1.5 × 10\^9 /L * Hemoglobin ≥ 9 g/dL * Platelets ≥ 75 × 10\^9 /L * Aspartate aminotransferase (AST/SGOT) and alanine aminotransferase (ALT/SGPT) ≤ 2.5 × upper limit of normal (ULN) or ≤ 5.0 × ULN if liver metastases are present * Total bilirubin ≤ 1.5 × ULN; or \< 2.5 × ULN if Gilbert syndrome or disease involving liver * Creatinine clearance \>30 mL/min (Cockcroft-Gault formula) * Adequate blood coagulation function as evidence by an international normalized ratio (INR) ≤ 1.5 unless participant is on anticoagulants * Tumor biopsy during screening is required if safe and feasible. If archival tissue is available from a previous biopsy performed within 90 days of screening assessments, a repeat screening biopsy is not required even if safe and feasible. If neither option is possible, archival tissue from any timepoint should be requested, if available. Exclusion Criteria: * The presence of any of the following criteria excludes a patient from participating in the study: * Pure small cell carcinoma * Previous exposure to multi-TKI therapies. * Uncontrolled hypertension (persistent systolic blood pressure ≥ 140 mm Hg and/or diastolic blood pressure ≥ 90 mm Hg) or known coronary artery disease with angina. Patients with known hypertension must be on antihypertensive medication with BPs generally \<140/90 to be eligible. * History of congestive heart failure of Class II-IV New York Heart Association criteria, or serious cardiac arrhythmia requiring treatment (except atrial fibrillation, paroxysmal supraventricular tachycardia), history of myocardial infarction within 6 months of study entry, or prolongation of QTc interval to \>480 msec using Fridericia formula (QTcF) at screening (except for participants with pacemakers, where there is no QTc cutoff). * Any prior or concurrent malignancy whose natural history or treatment has the potential to interfere with the safety or efficacy assessments. * Symptomatic and/or untreated CNS metastases. * Pre-existing duodenal stent or any gastrointestinal disorder or defect which would interfere with absorption of study medication, as determined by the Investigator. * Persistent requirement for corticosteroids at equivalent of \>10 mg QD prednisone within 14 days before study treatment start. * Other anticancer therapies within 3 weeks of study treatment start, or within 5 half-lives of study treatment start for non-cytotoxic oral agents, whichever is shorter; with the exception of androgen deprivation therapy, enzalutamide, or abiraterone acetate which should be continued through study treatment. * Palliative radiation within 2 weeks of study treatment start.