Key Inclusion Criteria:  Female and at least 18 years old. Histologically or cytologically confirmed recurrent high grade epithelial ovarian, primary peritoneal, or fallopian tube cancer with histologic subtypes of serous, clear cell or endometrial for which there is no known or established treatment available with curative intent. Have demonstrated relapse within 6 months of platinum therapy (platinum-free interval <6 months). Must have evaluable or measurable disease according to RECIST v1.1 criterion: defined as at least one lesion that can be accurately measured. Adequate hematologic and organ function. Ability and willingness to take oral medication. If unknown homologous recombination deficiency (HRD) status, subjects must provide formalin-fixed, paraffin-embedded tumor samples available from the primary or recurrent cancer or agree to undergo fresh biopsy prior to study treatment initiation.  Willingness to release archival tissue for research purposes or to undergo a tumor tissue biopsy prior to dosing on Cycle 1 Day 1.  Additional Key Inclusion Criteria for Phase II:  This supersedes inclusion criterion 3 (above). Demonstrated relapse within 6 months of platinum therapy, while taking a PARPi as maintenance: a minimum of 3 months is required if the participant received PARPi maintenance following first-line chemotherapy. Must have measurable disease according to RECIST V1.1 criterion: defined as at least one lesion that can be accurately measured.  Key Exclusion Criteria:  Prior therapy directed at the malignant tumor within the last four weeks prior to Cycle 1 Day 1 (6 weeks for nitrosoureas or mitomycin C). A minimum of 10 days between termination of the prior PARPi and administration of ZN-c3 and niraparib treatment is required. Any investigational drug therapy <28 days. Prior treatment with a WEE1 inhibitor. Known hypersensitivity to any drugs similar to ZN-c3 and/or niraparib in class or its excipients. Participant has any known history or current diagnosis of myelodysplastic syndrome (MDS) or acute myeloid leukemia (AML). Uncontrolled hypertension (Diastolic BP > 90 mmHg or Systolic BP > 140 mmHg). Myocardial impairment of any cause (e.g., cardiomyopathy, ischemic heart disease, significant valvular dysfunction, hypertensive heart disease, and congestive heart failure) resulting in heart failure by New York Heart Association Criteria (Class III or IV). Significant gastrointestinal abnormalities, requirement for IV alimentation, active peptic ulcer, chronic diarrhea, or vomiting considered to be clinically significant in the judgment of the Investigator, or prior surgical procedures affecting absorption. 12-lead ECG demonstrating a corrected QT interval using Fridericia's formula (QTcF) of >480 ms, except for subjects with atrioventricular pacemakers or other conditions (e.g., right bundle branch block) that render the QT measurement invalid. History or current evidence of congenital or family history of long QT syndrome or Torsades de Pointes (TdP). Taking medications with a known risk of TdP (according to current information provided at https://crediblemeds.org).