Combination Immunotherapy in Subjects With Advanced HPV Associated Malignancies

Background: More than 30,000 cases of human papillomavirus (HPV) associated cancers occur annually in the United States. When these cancers spread, they do not respond well to standard treatments and are often incurable. Researchers want to see if a mix of drugs can help. Objective: To learn if a mix of immunotherapy drugs can shrink tumors in people with HPV associated cancers. Eligibility: People ages 18 and older with locally advanced or metastatic HPV associated cancer, such as cervical cancers; cyclin-dependent kinase inhibitor 2A (P16+) oropharyngeal cancers; anal cancers; vulvar, vaginal, penile, and squamous cell rectal cancers; or other locally advanced or metastatic solid tumors (e.g., lung, esophagus) that are known HPV+ cancers Design: Participants will be screened with: * medical history * disease confirmation (or tumor biopsy) * physical exam * body scans (computed tomography (CT), magnetic resonance imaging (MRI), and/or nuclear) * blood tests * electrocardiogram (to measure the electrical activity of the heart) * urine tests. Participants will get PDS0101 injected under the skin every 4 weeks for 6 doses. Then they will get it every 3 months for 2 doses. Participants will get M7824 (MSB0011395C) by intravenous infusion every 2 weeks. For this, a needle is inserted into a vein. The drug is given over a 1-hour period. Participants will get NHS-IL12 injected under the skin every 4 weeks. Participants will get the study drugs for up to 1 year. They will visit the NIH every 2 weeks. They will repeat the screening tests during the study. About 28 days after treatment ends, participants will have a follow-up visit or telephone call. Then they will be contacted every 3 months for 1 year, and then every 6 months after that, for the rest of their life. Patients with cervical cancer with prior pelvic radiation and boost brachytherapy will be enrolled in a separate cohort to evaluate safety and preliminary evidence of efficacy...

* INCLUSION CRITERIA: Subjects with cytologically or histologically confirmed locally advanced or metastatic human papilloma virus (HPV) associated malignancies: * Cervical cancers; * cyclin-dependent kinase inhibitor 2A (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 must have measurable disease, per response evaluation criteria in solid tumors (RECIST) 1.1. Subjects must have received one prior line of systemic chemotherapy as well as checkpoint therapy if checkpoint therapy is Food and Drug Administration (FDA) approved for that specific tumor type (e.g., head and neck squamous cell carcinoma (HNSCC) and programmed death-ligand 1 (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. Age \>= 18 years. Eastern Cooperative Oncology Group (ECOG) performance status \=1 x 10\^9/L; * Hemoglobin \>= 9 g/dL; * Platelets \>=75,000/microliter. Adequate renal and hepatic function at screening, as follows: * Serum creatinine \=40 mL/min for participant with creatinine levels \> 1.5 X institutional ULN (GFR can also be used in place of creatinine or CrCl); * Bilirubin \