Pilot Trial of a DNA Vaccine Encoding Prostatic Acid Phosphatase (pTVG-HP) and PD-1 Blockade, With Targeted Ablation of Resistant Lesions, in Patients With Non-Castrate Recurrent Oligometastatic Prostate Cancer

Inclusion Criteria: * Participants must be at least 18 years of age with a histologic diagnosis of adenocarcinoma of the prostate * Participants must have undergone radical prostatectomy * Participants must have completed local therapy by surgery, and any adjuvant/salvage radiation therapy (if required), at least 3 months prior to entry, with removal or ablation of all visible disease, including seminal vesical and/or local lymph node involvement. * Participants must have biochemically recurrent disease defined by the following: * PSA doubling time, calculated from most recent 4 serum PSA values (collected up to one year prior to enrollment, at least 2 weeks apart, and all from the same clinical laboratory), must be a positive number (i.e. evidence of PSA rise over time). * Participants must have oligometastatic disease, defined as: * \< 3 lesions consistent with metastases as detected by CT of the abdomen/pelvis and bone scintigraphy (bone scan) * Lesions consistent with metastatic prostate cancer as detected by PSMA PET/CT * Participants with a prior history of a second malignancy are eligible provided they have been treated with curative intent and have been free of disease greater than three years. There will be no exclusion for patients with a history of basal cell carcinoma, squamous cell skin cancer, superficial bladder cancer, or other in situ carcinoma that has been adequately treated. * Participants who are sexually active must use a reliable form of contraception while on study and for 4 weeks after the last immunization. * ECOG performance score \< 2 and life expectancy of at least 12 months. * Participants must have normal hematologic, renal and liver function * Participants must be informed of the experimental nature of the study and its potential risks and must sign an IRB-approved written informed consent form indicating such an understanding. * Willingness to provide blood samples for immune studies, per study calendar, up to one year after study, even if off treatment. Exclusion Criteria: * Small cell or other variant prostate cancer histology * Participants cannot have evidence of immunosuppression or have been treated with immunosuppressive therapy, such as chemotherapy or chronic treatment dose corticosteroids (greater than the equivalent of 10 mg prednisone per day), within 3 months of the first vaccination. * Seropositive for HIV, hepatitis B (HBV) or hepatitis C (HCV) per patient history due to the immunosuppressive features of these diseases. * Prior treatment with an LHRH agonist or nonsteroidal antiandrogen, except in the following circumstances: Neoadjuvant/adjuvant androgen deprivation therapy administered with radiation therapy or at the time of prostatectomy is acceptable, provided that there was no evidence of PSA progression while on treatment. In this situation, patients must not have received more than 24 months of androgen deprivation treatment. Other treatment with androgen deprivation therapy is prohibited. * Serum testosterone at screening \< 50 ng/dL. * Participants must not be concurrently taking other medications or supplements with known hormonal effects, including PC-SPES, megestrol acetate, finasteride, ketoconazole, estradiol, or Saw Palmetto. All other medications with possible anti-cancer effects must be discussed with the PI prior to study entry. * Participants previously treated with other potential or experimental therapies for prostate cancer must have discontinued these treatments and completed at least a 4 week washout prior to beginning treatment. * Participants must not have known psychological or sociological conditions, addictive disorders or family problems, which would preclude compliance with the protocol. * Participants with unstable or severe intercurrent medical conditions or laboratory abnormalities that would impart, in the judgment of the PI, excess risk associated with study participation or study agent administration. * Participants who have concurrent enrollment on other phase I, II, or III investigational therapeutic treatment studies for prostate cancer cannot be actively receiving treatment and the last dose cannot be within 4 weeks of day 1. They must be in the follow up phase of the study.