A Phase I/II Study of Combination Immunotherapy for Advanced Cancers Including HPV-Associated Malignancies, Small Bowel, and Colon Cancers

Background: Often, metastatic human papillomavirus (HPV) associated cancers cannot be cured. They also do not respond well to treatment. Some forms of colon cancer also have poor responses to treatment. Researchers want to see if a new drug treatment can help people with these types of cancers. Objective: To find a safe dose of entinostat in combination with PDS01ADC and bintrafusp alfa and to see if this treatment will cause tumors to shrink. Eligibility: Adults ages 18 and older who have cervical, oropharyngeal, anal, vulvar, vaginal, penile, squamous cell rectal, or another cancer that may be associated with HPV infection or microsatellite stable small bowel or colorectal cancer. Design: Participants will be screened with a medical history and physical exam. Their ability to do daily activities will be assessed. They may have imaging scans of the brain and/or chest, abdomen, and pelvis. They may have nuclear bone scans. They will have an electrocardiogram to test heart function. They will have blood and urine tests. They may have a tumor biopsy. Participants with skin lesions may have them photographed. Some screening tests will be repeated during the study. Treatment will be done in 28-day cycles. Participants will get bintrafusp alfa through an intravenous catheter every 2 weeks. They will get PDS01ADC as an injection under the skin every 4 weeks. They will take entinostat by mouth once a week. They will complete a medicine diary. Participants will get treatment for 2 years. They will have 1-2 follow-up visits in the 30 days after treatment ends. Then they will be contacted every 6 months to check on their health.

* INCLUSION CRITERIA: * Phase I: Subjects with cytologically or histologically confirmed locally advanced or metastatic HPV associated malignancies or MSS small bowel or colorectal cancer (Cohort 1). * Phase II: Subjects with cytologically or histologically confirmed locally advanced or metastatic HPV associated malignancies (Cohort 2), or MSS small bowel or colorectal cancer (Cohort 3). * Cohort 2 includes: * Cervical cancers; * P16+ Oropharyngeal cancers; * Anal cancers; * Vulvar, vaginal, penile, and squamous cell rectal cancers; * Other locally advanced or metastatic solid tumors (e.g., lung, esophagus) that are known HPV+. * Subjects with MSS colorectal or small bowel cancer must have received two prior lines of systemic chemotherapy. Subjects with HPV associated malignancies must have received one prior line of systemic chemotherapy as well as checkpoint therapy if checkpoint therapy is FDA approved for that specific tumor type (e.g., HNSCC and PDL1+ cervical cancer). Prior checkpoint therapy is not needed where checkpoint therapy has not been FDA approved for that specific tumor type (e.g., anal, vaginal, vulvar, penile, PDL1 negative cervical). Exceptions to the above include participant who are not eligible to receive the above therapies or who decline these standard treatment options after appropriate counseling has been provided. * Subjects must have measurable disease, per RECIST 1.1. * Age \>=18 years. * ECOG performance status \=1.5 x 10\^9/L; * Hemoglobin \>= 9 g/dL; * Platelets \>= 100,000/microliter. * Adequate renal and hepatic function at screening, as follows: * Measured or calculated creatinine clearance (using the Cockcroft-Gault equation) \> 50 mL/min * Bilirubin \