Pomalidomide and Nivolumab in People With Virus-Associated Malignancies With or Without HIV

* INCLUSION CRITERIA: * Histologically or cytologically proven selected virus-associated tumors that are systemic, metastatic or locally advanced and not amenable to curative treatment options or are relapsed/refractory to first-line therapy as appropriate for each tumor type as outlined below. Also, participants with eligible solid tumors, who after evaluation by an expert in the area are deemed to be potentially curable after extensive surgery, but refuse such surgical procedure due to associated disfigurement and/or morbidity, may be eligible for the study with the necessary informed consent. Pathology confirmation by NCI Laboratory of Pathology is needed for eligibility. The following tumor types listed below are eligible, and require assessing of virus infection of the tumor cells with EBV EBER by in situ hybridization (ISH), KSHV LANA , p16, and Merkel cell polyomavirus large T antigen by immunohistochemistry (IHC) to document the respective viral infection (EBV, KSHV, HPV, MCPvY); or detection of serum HBV surface antigen, anti-HBV core antibody, elevated HBV DNA viral load, positive HVC antibody or elevated HCV RNA viral load. The tumor types studied in the phase 1 trial will be as below. For tumors where \>95% are known to be virus-associated, such as cervical cancer, confirmation of virus status is not required. --EBV-positive Hodgkin lymphoma meeting the following criteria: * Relapsed or refractory de novo classical Hodgkin lymphoma having failed standard first-line therapy; and * Unresponsive or progressive disease after treatment with brentuximab vedotin or may be brentuximab vedotin na(SqrRoot) ve but is ineligible or unable to receive brentuximab vedotin; and * Unresponsive or progressive disease after checkpoint inhibitor therapy; and * Unresponsive or progressive disease after or is ineligible for autologous stem cell transplant (auto-SCT) --EBV-positive aggressive non-Hodgkin lymphomas meeting the following criteria: * Relapsed/refractory disease after standard first-line chemotherapy; and * Relapsed disease after autologous stem cell transplant if indicated for histology (i.e diffuse large B-cell lymphoma relapsed more than one year after first line treatment) or autologous stem cell transplant is not feasible; and * Relapsed after CAR-T cell therapy for HIV-negative participants only if indicated for histology (i.e., diffuse large B-cell lymphoma) or CAR-T cell therapy is not feasible * EBV-positive nasopharyngeal cancer unresponsive or progressive disease on or after platinum-containing chemotherapy and/or radiotherapy * EBV-positive gastric cancer that is unresponsive or progressive disease on or after first-line chemotherapy * EBV-positive leiomyosarcomas that is unresponsive or progressive disease on or after 2 systemic regimens (CCRT/platinum-taxane) * Kaposi sarcoma impairing physical wellbeing (for example, tumor edema, pain, skin ulceration or breakdown, oral disease impairing function), no active KSHV-associated multicentric Castleman disease in past 12 months, and one or more of the following: * Inadequate tumor response after 6 or more cycles of liposomal doxorubicin or paclitaxel or other active cytotoxic agents (i.e. etoposide, bleomycin, anthracyclines, vincristine, vinblastine); or * Progressive disease while receiving liposomal doxorubicin or paclitaxel or other active cytotoxic agents (i.e. etoposide, bleomycin, anthracyclines, vincristine, vinblastine); or * Intolerant of liposomal doxorubicin and paclitaxel * Primary effusion lymphoma unresponsive or progressive disease on or after first-line combination chemotherapy * HPV-positive head and neck cancer that is unresponsive or progressive on or after first-line combination chemotherapy +/- radiotherapy * HPV-positive cervical cancer that is unresponsive or progressive on at least one systemic regimen for recurrent (does not include initial CCRT) or metastatic disease. Tumor HPV testing will not be a requirement for study eligibility for cervical cancer. * HPV-positive anal cancer that is unresponsive or progressive on or after first-line combination chemotherapy +/- radiotherapy * HPV-positive vaginal cancer that is unresponsive or progressive on or after first-line chemotherapy * HPV-positive penile cancer that is unresponsive or progressive on or after surgery and first-line chemotherapy * HPV-positive vulvar cancer that is unresponsive or progressive on or after first- line combination chemotherapy * MCPyV-positive Merkel cell carcinomas that is relapsed or refractory after prior checkpoint inhibitor therapy * HBV- or HCV-associated hepatocellular carcinoma that is not amenable to local therapy or liver transplant and has progressed on first-line therapy with sorafenib or levatinib or atezolizumab+bevacizumab * For solid tumors, participants must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>=20 mm (\>=2 cm) by chest x-ray or as \>=10 mm (\>=1 cm) with CT scan, MRI, or calipers by clinical exam. * For hematologic malignancies, participants must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (lymph nodes must measure \>=15 mm in the short axis and extranodal lesions must measure \>=10 mm in the short axis with CT scan. For primary effusion lymphoma, body cavity effusions may be followed as measurable disease by CT scan. * For KS, participants must have measurable disease per the modified ACTG criteria, defined as at least five measurable cutaneous KS lesions with no previous local radiation, surgical or intralesional cytotoxic therapy that would prevent response assessment for that lesion or in the event that participants have assessable disease, response will be evaluated per RECIST criteria per solid tumor guidelines. *