Enzalutamide and PDS01ADC in PET Positive Recurrent Prostate Cancer (pprPC) Without Testosterone Lowering Therapy

* Enzalutamide given with androgen deprivation therapy (ADT) is Food and Drug Administration (FDA) approved for the treatment of metastatic prostate cancer based on conventional computed tomography (CT) and Tc99 scan. * Enzalutamide for 3 months (short course) given without ADT has demonstrated the ability to control prostate-specific antigen (PSA) in recurrent prostate cancer for nearly a year, delaying the need for additional therapy. * Enzalutamide without ADT was very well tolerated in our previous study, a prerequisite for any therapy in recurrent disease where patients may not have symptoms from prostate cancer for 5-10 years. * Enzalutamide without ADT demonstrated the ability to enhance natural killer (NK) cells and decrease myeloid-derived suppressor cells. * PDS01ADC is an immunocytokine that binds to areas of necrotic tumors. Preclinical data have demonstrated that PDS01ADC delivery to the tumor is enhanced by cytotoxic therapies such as radiation and chemotherapy. * PDS01ADC has been shown to be well tolerated and even induce PSA responses in patients with recurrent prostate cancer. * PDS01ADC has also been able to enhance NK cells in prostate cancer patients. * Higher levels of NK cells have been associated with better clinical outcomes in prostate cancer. * Prostate-specific membrane antigen (PSMA) positron emission tomography (PET) imaging is now approved in recurrent prostate cancer. No trial has prospectively evaluated an anti-androgen therapy (e.g., enzalutamide) without ADT in this population. If changes in imaging are seen similar to the PSA responses noted previously, these findings may demonstrate the efficacy of enzalutamide-based regimens in recurrent prostate cancer. * Given that enzalutamide is cytotoxic and will induce necrosis, there is a rationale to combine it with the necrosis-targeting agent PDS01ADC. The fact that both enhance NK cells, which have been associated with better clinical outcomes adds further rationale to this combination. Objective: -To determine if the combination of enzalutamide and immunotherapy (PDS01ADC) is associated with an increase in the duration of PSA suppression compared to that of enzalutamide alone in participants with PET Positive Recurrent Prostate Cancer (pprPC). Eligibility: * Participant must provide documentation of histologic or cytological confirmation of prostate cancer or tumor sample for diagnosis confirmation. Note: in the absence of pathology or documentation, participant must have a rising PSA, PSMA plus disease, and his history consistent with prostate cancer as documented by the investigator. * History of primary treatment for prostate cancer (either surgery or radiation). * PSA doubling time within less than 1 year before treatment initiation. * Testosterone \>100 ng/dL. * Age \>=18 years. * Evidence of prostate cancer on PSMA PET/CT scan. * Negative Tc99 Bone Scan. * No evidence of soft tissue disease on the CT scan (or MRI) per the Response Evaluation Criteria in Solid Tumors (RECIST) guideline (version 1.1). Design: * This is an open-label phase II clinical trial with two treatment arms: Arm 1 (enzalutamide) and Arm 2 (enzalutamide plus PDS01ADC). * After enrollment, participants will be randomized between Arms 1 and 2 and receive 3 cycles of treatment of enzalutamide or enzalutamide and PDS01ADC. * During off treatment monitoring period following the third cycle, participants who experience PSA recovery to baseline will have a second course of enzalutamide treatment only (3 cycles total).