Study of Pembrolizumab and Lenvatinib in Metastatic and Recurrent Cervix Cancer (LenPem Cervix)

Inclusion Criteria: Participants are eligible to be included in the study only if all the following criteria apply: 1. Female participants who are at least 18 years of age on the day of signing informed consent. 2. Histologically confirmed diagnosis of squamous, adenocarcinoma or adenosquamous cervical cancer, that is recurrent or metastatic. 3. Prior therapy: May have received up to 2 prior lines of systemic chemotherapy in the setting of advanced, metastatic (Stage IVB) or recurrent cervical cancer. May have received prior checkpoint inhibitor for advanced, metastatic (Stage IVB) or recurrent cervical cancer. May have received prior bevacizumab or antiangiogenic agent for recurrent or metastatic cervical cancer, 4. Include whether prior checkpoint inhibitor was used in first line setting or second line setting. 5. Prior Radiation therapy will be allowed and not counted as a line of treatment. 6. Prior chemotherapy used as radiation sensitizer (e.g. cisplatin) used as treatment during chemoradiation will be allowed and counted as a line of treatment. 7. Female participants: * A female participant is eligible to participate if she is not pregnant, not breastfeeding, and at least one of the following conditions applies: * Not a woman of childbearing potential (WOCBP) OR Is a WOCBP and using a contraceptive method that is highly effective (with a failure rate of \<1% per year), with low user dependency, or be abstinent from heterosexual intercourse as their preferred and usual lifestyle (abstinent on a long term and persistent basis), during the intervention period and for at least 120 days post pembrolizumab or 30 days post lenvatinib whichever occurs last. The investigator should evaluate the potential for contraceptive method failure (i.e., noncompliance, recently initiated) in relationship to the first dose of study intervention. * A WOCBP must have a negative highly sensitive pregnancy test (urine or serum as required by local regulations) within 24 hours before the first dose of study intervention. * If a urine test cannot be confirmed as negative (e.g., an ambiguous result), a serum pregnancy test is required. In such cases, the participant must be excluded from participation if the serum pregnancy result is positive. * The investigator is responsible for review of medical history, menstrual history, and recent sexual activity to decrease the risk for inclusion of a woman with an early undetected pregnancy. 8. Participants must have a PD-L1 diagnostic test of primary or recurrent archival tumor tissue. 9. Participants may have progressed on treatment with an anti-PD-1/L1 mAb administered either as monotherapy or in combination with other checkpoint inhibitors or other therapies. PD-1 treatment progression is defined by meeting all the following criteria: 1. Has received at least 2 doses of an approved anti-PD-1/L1 mAb. 2. Has demonstrated disease progression after anti-PD-1/L1 as defined by RECIST v1.1. The initial evidence of PD is to be confirmed by a second assessment no less than 4 weeks from the date of the first documented disease progression, in the absence of rapid clinical progression. 3. Progressive disease has been documented within 12 weeks from the last dose of anti-PD-1/L1 mAb. i. Progressive disease is determined according to iRECIST. ii. This determination is made by the investigator. Once disease progression is confirmed, the initial date of disease progression documentation will be considered the date of disease progression. 10. Participants who have AEs due to previous anticancer therapies must have recovered to ≤Grade 1 or baseline. Participants with endocrine-related AEs who are adequately treated with hormone replacement or participants who have ≤Grade 2 neuropathy are eligible. 11. The participant (or legally acceptable representative if applicable) provides written informed consent for the trial. 12. Have measurable disease based on RECIST 1.1. Lesions situated in a previously irradiated area are considered measurable if progression has been demonstrated in such lesions. 13. Archival tumor tissue sample or newly obtained \[core, incisional or excisional\] biopsy of a tumor lesion not previously irradiated has been provided. Formalin-fixed, paraffin embedded (FFPE) tissue blocks are preferred to slides. Newly obtained biopsies are preferred to archived tissue. 14. Have an Eastern Cooperative Oncology Group performance status of 0 to 1. Evaluation of ECOG is to be performed within 7 days prior to the first dose of study intervention. 15. Have adequate organ and marrow function as defined in the following table (Table 2). Specimens must be collected within 10 days prior to the start of study intervention. 16. Criteria for known Hepatitis B and C positive subjects. Hepatitis B and C screening tests are not required unless: * Known history of HBV or HCV infection * As mandated by local health authority 17. Hepatitis B positive subjects * Participants who are HBsAg positive are eligible if they have received HBV antiviral therapy for at least 4 weeks and have undetectable HBV viral load prior to randomization. * Participants should remain on anti-viral therapy throughout study intervention and follow local guidelines for HBV anti-viral therapy post completion of study intervention. 18. Participants with history of HCV infection are eligible if HCV viral load is undetectable at screening. \- Participants must have completed curative anti-viral therapy at least 4 weeks prior to randomization. 19. Have adequately controlled BP with or without antihypertensive medications, defined as BP ≤150/90 mmHg with no change in antihypertensive medications within 1 week prior to randomization. 20. Ability to understand and the willingness to sign a written informed consent. Exclusion Criteria: Participants are excluded from the study if any of the following criteria apply: 1. A WOCBP who has a positive urine pregnancy test within 72 hours prior to enrollment. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required. Note: in the event that 72 hours have elapsed between the screening pregnancy test and the first dose of study treatment, another pregnancy test (urine or serum) must be performed and must be negative in order for subject to start receiving study medication. 2. Has received prior systemic anti-cancer therapy including investigational agents within 2 weeks prior to allocation. 3. Has received prior radiotherapy within 2 weeks of start of study intervention or radiation-related toxicities requiring corticosteroids. Note: 2 weeks or fewer of palliative radiotherapy for non-CNS disease, with a 1-week washout, is permitted. 4. Has received a live vaccine or live-attenuated vaccine within 30 days before the first dose of study intervention. Administration of killed vaccines is allowed. Note: please refer to Section 4.9 for information on COVID-19 vaccines. 5. Has received an investigational agent or has used an investigational device within 4 weeks prior to study intervention administration. 6. Has a diagnosis of immunodeficiency or is receiving chronic systemic steroid therapy (in dosing exceeding 10 mg daily of prednisone equivalent) or any other form of immunosuppressive therapy within seven days prior to the first dose of study drug. 7. Known additional malignancy that is progressing or has required active treatment within the past five years. Note: Participants with basal cell carcinoma of the skin, squamous cell carcinoma of the skin, or carcinoma in situ, excluding carcinoma in situ of the bladder, that have undergone potentially curative therapy are not excluded. 8. Has known active CNS metastases and/or carcinomatous meningitis. Participants with previously treated brain metastases may participate provided they are radiologically stable, i.e. without evidence of progression for at least 4 weeks by repeat imaging (note that the repeat imaging should be performed during study screening), clinically stable and without requirement of steroid treatment for at least 14 days prior to first dose of study intervention. 9. Has severe hypersensitivity (≥Grade 3) to pembrolizumab and/or any of its excipients. 10. Has active autoimmune disease that has required systemic treatment in the past 2 years except replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid) 11. Has a history of (non-infectious) pneumonitis/interstitial lung disease that required steroids or has current pneumonitis/interstitial lung disease. 12. Has an active infection requiring systemic therapy. 13. Has a known history of Human Immunodeficiency Virus (HIV) infection. Note: No HIV testing is required unless mandated by local health authority. 14. Concurrent active Hepatitis B (defined as HBsAg positive and/or detectable HBV DNA) and Hepatitis C virus (defined as anti-HCV Ab positive and detectable HCV RNA) infection. Note: Hepatitis B and C screening tests are not required unless: * Known history of HBV and HCV infection * As mandated by local health authority 15. Has had major surgery within three weeks prior to first dose of study interventions. Note: Adequate wound healing after major surgery must be assessed clinically, independent of time elapsed for eligibility. 16. Has a history or current evidence of any condition, therapy, or laboratory abnormality or other circumstance that might confound the results of the study, interfere with the participant's participation for the full duration of the study, such that it is not in the best interest of the participant to participate, in the opinion of the treating investigator. 17. Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial. 18. Is pregnant or breastfeeding or expecting to conceive or father children within the projected duration of the study, starting with the screening visit through 120 days after the last dose of trial treatment. 19. Has had an allogenic tissue/solid organ transplant. 20. Has preexisting ≥Grade 3 gastrointestinal or non-gastrointestinal fistula. 21. Has urine protein ≥1 g/24 hours. Note: Participants with proteinuria ≥2+ (≥100 mg/dL) on urinalysis will undergo 24-hour urine collection for quantitative assessment of proteinuria. 22. Has a LVEF below the institutional (or local laboratory) normal range, as determined by multigated acquisition (MUGA) or echocardiogram (ECHO). 23. Has radiographic evidence of encasement or invasion of a major blood vessel, or of intratumoral cavitation. Note: The degree of proximity to major blood vessels should be considered because of the potential risk of severe hemorrhage associated with tumor shrinkage/necrosis following lenvatinib therapy 24. Prolongation of QTcF interval to \>480 ms. 25. Has clinically significant cardiovascular disease within 12 months from first dose of study intervention, including New York Heart Association Class III or IV congestive heart failure, unstable angina, myocardial infarction, cerebral vascular accident, or cardiac arrhythmia associated with hemodynamic instability. Note: Medically controlled arrhythmia would be permitted. 26. Gastrointestinal malabsorption or any other condition that might affect the absorption of Lenvatinib. 27. Active hemoptysis (bright red blood of at least 0.5 teaspoon) within three weeks prior to the first dose of study drug.