BiCaZO: A Study Combining Two Immunotherapies (Cabozantinib and Nivolumab) to Treat Patients With Advanced Melanoma or Squamous Cell Head and Neck Cancer, an immunoMATCH Pilot Study

PRIMARY OBJECTIVES: I. To evaluate the feasibility of molecular characterization based on tumor mutation burden (TMB) for participant stratification, as assessed by the proportion of participants with less than or equal to a 21-day turnaround time for biopsy results in Stage I of the study. II. To evaluate the feasibility of molecular characterization based on TMB and Nanostring IO360 gene expression profiling assay for tumor inflammation score (TIS) for stratification, as assessed by the proportion of participants with less than or equal to a 21-day turnaround time for biopsy results in Stage II of the study. III. To evaluate the efficacy by overall response rate (ORR - defined as confirmed and unconfirmed partial responses plus complete responses) of cabozantinib S-malate (cabozantinib) plus nivolumab in each disease cohort, stratified by TIS. SECONDARY OBJECTIVES: I. To assess the difference in ORR in each disease cohort between tumor marker subgroups separately for each disease cohort. II. To assess safety and tolerability of this treatment in these populations. III. To estimate disease control rate (DCR) in participants receiving cabozantinib plus nivolumab in each disease cohort, stratified by tumor biomarkers. IV. To estimate progression-free survival (PFS) in participants receiving cabozantinib plus nivolumab in each disease cohort, stratified by tumor biomarkers. V. To estimate overall survival (OS) in participants receiving cabozantinib plus nivolumab in each disease cohort, stratified by tumor biomarkers. VI. To assess the proportion of participants with assay failure, and the time from the date of tissue collection to molecular group determination at the end of Stage I. VII. To evaluate the ORR in participants receiving cabozantinib plus nivolumab in each disease cohort, stratified by TIS and TMB. VIII. To evaluate the association of TMB as a continuous variable with ORR within each disease cohort. ADDITIONAL OBJECTIVE: I. To identify candidate biomarkers predictive of response and toxicity to the cabozantinib and nivolumab combination. BANKING OBJECTIVE: I. To bank specimens for future correlative studies. OUTLINE: Patients receive nivolumab intravenously (IV) over 30 minutes on day 1 of each cycle and cabozantinib orally (PO) daily. Cycles repeat every 28 days for 2 years in the absence of disease progression or unacceptable toxicity. Patients also undergo computed tomography (CT) or magnetic resonance imaging (MRI) scans and collection of blood samples throughout the trial. Patients undergo a tumor biopsy during screening and optionally during follow-up. After completion of study treatment, patients are followed up every 12 weeks for 1 year and then every 6 months until 3 years after step 2 registration.