Inclusion Criteria:  One of the following histologically or cytologically confirmed, advanced (locally recurrent unresectable or metastatic) solid tumors:  Squamous cell carcinoma, adenosquamous carcinoma, or adenocarcinoma of the cervix Endometrial cancer Head and neck swuamous cell carcinoma (HNSCC) Unresectable biliary adenocarcinoma (gallbladder or biliary tree [intrahepatic or extrahepatic] cholangiocarcinoma) Adenocarcinoma or squamous cell carcinoma of the esophagus or advanced/metastatic Siewert type 1 adenocarcinoma of the gastroesophageal junction (GEJ). Triple-negative breast cancer (TNBC) Hepatocellular carcinoma (HCC) Measurable disease per RECIST v1.1 as assessed by BICR or local site investigator. Adequately controlled blood pressure (BP) with or without antihypertensive medications. Human immunodeficiency virus (HIV)-infected participants must have well controlled HIV on anti-retroviral therapy (ART). Male participants must agree to follow contraceptive guidance. Female participants are not pregnant or breastfeeding, not a woman of child-bearing potential (WOCBP) or is a WOCBP and agrees to follow contraceptive guidance. Adequate organ function.  Exclusion Criteria:  History of a second malignancy, unless potentially curative treatment has been completed with no evidence of malignancy for 3 years. Prior therapy with anti-programmed cell-death (PD-1), anti-PD-L1, anti-PD-L2, or anti-T-cell immunoreceptor with Ig and ITIM domains (TIGIT) agent. Prior systemic anticancer therapy including investigational agents within 4 weeks before randomization/allocation. Received a live or live-attenuated vaccine within 30 days before the first dose of study intervention. Administration of killed vaccines are allowed. Diagnosis of immunodeficiency or is receiving chronic systemic steroid therapy or any other form of immunosuppressive therapy within 7 days before the first dose of study medication. Active autoimmune disease that has required systemic treatment in past 2 years. Active infection requiring systemic therapy. Concurrent active hepatitis B and hepatitis C virus infection. History of allogenic tissue/solid organ transplant. Previous treatment with lenvatinib (for participants who will receive lenvatinib in their assigned treatment arm). History of congestive heart failure or myocardial infarction within 6 months of treatment (for participants who will receive paclitaxel in their assigned allocation arm).