Pembrolizumab and Lenvatinib in Clear Cell Ovarian Cancer

This is an open label, non-randomized, single cohort phase II trial. Participants with recurrent or persistent clear cell ovarian carcinoma (CCOC) will be treated with the investigational combination of pembrolizumab and lenvatinib until progression of disease or unacceptable toxicity. The names of the study drugs involved in this study are: * Lenvatinib * Pembrolizumab The research study procedures include screening for eligibility and study treatment including evaluations and follow up visits. Participants will receive study treatment until progression of disease or unacceptable toxicity. Participants will be followed for 36 months thereafter. It is expected that about 31 people will take part in this research study. This research study is a Phase II clinical trial. Phase II clinical trials test the safety and effectiveness of an investigational drug to learn whether the drugs work in treating a specific disease. "Investigational" means that the drug is being studied. The U.S. Food and Drug Administration (FDA) has not approved lenvatinib or pembrolizumab for your specific disease but it has been approved for other uses.