The purpose of this research study is to test the safety of Niraparib and dostarlimab as a combination treatment and see what effects (good and bad) this combination treatment has on patients with recurrent or progressive cervix cancer.

Patients will have tests and exams to see if they are eligible for the clinical trial. If found eligible, the patient will receive treatment with Niraparib daily and dostarlimab by vein every three weeks for 4 cycles then every six weeks.

Patients will receive the study treatment as long as there is evidence that the tumor is not growing or spreading and they are not having any unacceptable, bad side effects.

Patients will be monitored during treatment with tests and exams and after treatment completion for up to 5 years.

Inclusion Criteria:

1. Patient is female at least 18 years of age.
2. Patient has histologically proven cervical cancer, which is recurrent or progressive
3. Patient has archival tumor tissue available or a fresh biopsy of recurrent or persistent tumor must be obtained prior to study treatment initiation. Availability of tissue does not affect eligibility of patient on trial, but PD-L1 status and next-generation sequencing data is required to be collected for the trial.
4. Patient has measurable lesions by RECIST v1.1.
5. Patient has an ECOG performance status of 0 to 1.
6. Patients must have received at least one or more prior systemic treatment regimens. Chemotherapy with radiation is not considered systemic treatment. Prior treatment with anti-PD-1, anti-PD-L1 or anti-PD-L2 therapies is allowed; however, these treatments could not have been discontinued due to immune related adverse events and patient cannot have progressed while on anti-PD-1, anti-PD-L1 or anti PD-L2 given in combination with chemotherapy or while on maintenance immunotherapy.
7. Patient has adequate organ function, defined per protocol.
8. Patient is able to take oral medications.
9. Participant must agree to not donate blood during the study or for 90 days after the last dose of study treatment.
10. If of childbearing potential, has a negative pregnancy test within 7 days prior to taking study medication or agrees to abstain from activities that could result in pregnancy from enrollment through 180 days after the last dose of study treatment, or be of non- childbearing potential.

Exclusion Criteria:

1. Known active central nervous system (CNS) metastases and/or carcinomatous meningitis. Note: Patients with previously treated brain metastases may participate provided they are stable for at least 4 weeks prior to the first dose of study treatment and have not been using steroids for at least 7 days prior to study treatment.
2. Known additional malignancy that required active treatment within the last 2 years. Exceptions include basal cell carcinoma of the skin or squamous cell carcinoma of the skin.
3. Patient is considered a poor medical risk that would interfere with cooperation with the requirements of the study.
4. Received a transfusion (platelets or red blood cells) ≤4 weeks prior to initiating protocol therapy.
5. Received colony stimulating factors (eg, granulocyte colony-stimulating factor, granulocyte macrophage colony stimulating factor, or recombinant erythropoietin) within 4 weeks prior initiating protocol therapy.
6. Known Grade 3 or 4 anemia, neutropenia or thrombocytopenia due to prior chemotherapy that persisted > 4 weeks and was related to the most recent treatment.
7. Known history of myelodysplastic syndrome (MDS) or acute myeloid leukemia (AML)
8. Serious, uncontrolled medical disorder, nonmalignant systemic disease, or active, uncontrolled infection. Examples include, but are not limited to, uncontrolled ventricular arrhythmia, recent (within 90 days) myocardial infarction, uncontrolled major seizure disorder, unstable spinal cord compression, superior vena cava syndrome, or any psychiatric disorder that prohibits obtaining informed consent
9. Pregnant or breastfeeding or expecting to conceive children within the projected duration of the study and for 180 days after the last dose of study treatment.
10. Diagnosis of immunodeficiency or is receiving systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to the first dose of study treatment. Participant receiving corticosteroids may continue as long as their dose is stable for least 4 weeks prior to initiating protocol therapy, and ≤ 10mg a day prednisone or equivalent.
11. Known history of human immunodeficiency virus (HIV) (HIV ½ antibodies).
12. Known active hepatitis B or hepatitis C.
13. Active autoimmune disease that has required systemic treatment in the past 2 years (i.e., with use of disease-modifying agents, corticosteroids, or immunosuppressive drugs). Replacement therapy (eg, thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment.
14. Not recovered to ≤Grade 1 or to baseline from chemotherapy induced AEs. Note: Patient with ≤ Grade 1 neuropathy or ≤ Grade 2 alopecia is an exception to this criterion and may qualify for the study.
15. Currently participating and receiving study therapy or has participated in a study of an investigational agent and received study therapy or used an investigational device within 4 weeks of the first dose of treatment.
16. Prior cytotoxic chemotherapy, anticancer targeted small molecules (e.g., tyrosine kinase inhibitors), hormonal agents within 5 half-lives, or monoclonal antibodies (mAb) within 5 half-lives or 4 weeks (whichever is shorter) of that treatment prior to study Day 1 or radiation therapy encompassing > 20% of the bone marrow within 2 weeks, 1 week for radiation therapy encompassing ≤ 20%.
17. Major surgery ≤ 3 weeks prior to initiating protocol therapy and participant must have recovered from any surgical effects.
18. Received a live vaccine within 14 days of planned start of study therapy.
19. Prior treatment with a known PARP inhibitor.
20. Known hypersensitivity to niraparib or dostarlimab components or excipients.
21. Patient experienced ≥ Grade 3 immune-related AE with prior immunotherapy, with the exception of non-clinically significant lab abnormalities: any immune-related AE (irAE) of Grade 3 or higher, immune-related severe neurologic events of any grade (e.g., myasthenic syndrome/myasthenia gravis, encephalitis, Guillain-Barré Syndrome, or transverse myelitis), exfoliative dermatitis of any grade (Stevens-Johnson Syndrome, toxic epidermal necrolysis, or drug reaction with eosinophilia and systemic symptoms [DRESS] syndrome), or myocarditis of any grade.
22. History of interstitial lung disease.