Lenvatinib (E7080/MK-7902) in Combination With Pembrolizumab (MK-3475) vs. Standard Chemotherapy and Lenvatinib Monotherapy in Participants With Recurrent/Metastatic Head and Neck Squamous Cell Carcinoma That Progressed After Platinum Therapy and Immunotherapy (MK-7902-009/E7080-G000-228/LEAP-009)

Inclusion Criteria: * Pathologically confirmed recurrent (not amenable to curative treatment with local and/or systemic therapies) or metastatic (disseminated) head and neck squamous cell carcinoma (HNSCC) of the oral cavity, oropharynx, hypopharynx, and/or larynx that is considered incurable by local therapies * Disease progression at any time during or after treatment with a platinum-containing (e.g., carboplatin or cisplatin) regimen * Disease progression on or after treatment with a programmed cell death protein 1/programmed death-ligand 1 monoclonal antibody (anti-PD-1/PD-L1 mAb) * Pre-study imaging that demonstrates evidence of disease progression based on investigator review of at least 2 pre-study images per RECIST 1.1, following initiation of treatment with a PD-1/PD-L1 inhibitor * Measurable disease by computed tomography scan (CT) or magnetic resonance imaging (MRI) based on Response Criteria in Solid Tumors Version 1.1 (RECIST 1.1) as verified by blinded independent central review (BICR). Tumor lesions situated in a previously irradiated area are considered measurable if progression has been demonstrated in such lesions * Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 assessed within 7 days of the first dose of study intervention * Male participants are eligible to participate if they agree to the following during the intervention period and for at least 1 week after the last dose of lenvatinib, 3 months after the last dose of capecitabine and paclitaxel, and 6 months after the last dose of docetaxel: * Refrain from donating sperm * Be abstinent from heterosexual intercourse as their preferred and usual lifestyle (abstinent on a long term and persistent basis) and agree to remain abstinent; or must agree to use contraception unless confirmed to be azoospermic * Contraceptive use by men should be consistent with local regulations regarding the methods of contraception for those participating in clinical studies * A female participant is eligible to participate if she is not pregnant or breastfeeding, and at least one of the following conditions applies: * Is not a woman of childbearing potential (WOCBP) * Is a WOCBP and using a contraceptive method that is highly effective (with a failure rate of \<1% per year), with low user dependency or be abstinent from heterosexual intercourse as their preferred and usual lifestyle (abstinent on a long term and persistent basis), during the intervention period and for at least 120 days post pembrolizumab or 1 month post lenvatinib, whichever occurs last (Arms 1 and 3), or during the intervention period and for at least 6 months after the last dose of capecitabine, docetaxel, paclitaxel; and 2 months after the last dose of cetuximab (Arm 2) * Female participants who randomize to Arm 2 must also agree not to donate or freeze/store eggs during the intervention period and for at least 6 months after the last dose of capecitabine, docetaxel, paclitaxel; and 2 months after the last dose of cetuximab * Adequately controlled blood pressure (BP) with or without antihypertensive medications * Participants who are hepatitis B surface antigen (HBsAg) positive are eligible if they have received hepatitis B virus (HBV) antiviral therapy for at least 4 weeks and have undetectable HBV viral load prior to randomization * Participants with history of hepatitis C virus (HCV) infection are eligible if HCV viral load is undetectable at screening * Adequate organ function Exclusion Criteria: * Disease that is suitable for local therapy administered with curative intent * Life expectancy of less than 3 months and/or has rapidly progressing disease in the opinion of the treating investigator * History of (noninfectious) pneumonitis/interstitial lung disease that required steroids, or has current pneumonitis/interstitial lung disease * Active infection requiring systemic therapy * Diagnosis of immunodeficiency or is receiving chronic systemic steroid therapy (in dosing exceeding 10 mg daily of prednisone equivalent) or any other form of immunosuppressive therapy within 7 days prior to the first dose of study drug * Known active central nervous system (CNS) metastases and/or carcinomatous meningitis * Known additional malignancy that is progressing or has required active systemic treatment within the past 3 years, except basal cell carcinoma of the skin, squamous cell carcinoma of the skin, superficial bladder cancer, or carcinoma in situ that have undergone potentially curative therapy * Active autoimmune disease that has required systemic treatment in the past 2 years * Had an allogeneic tissue/solid organ transplant * Known history of human immunodeficiency virus (HIV) infection * History of any contraindication or has a severe hypersensitivity to any components of pembrolizumab, lenvatinib or SOC chemotherapy. * Pre-existing ≥Grade 3 gastrointestinal or non-gastrointestinal fistula * History of a gastrointestinal malabsorption or any other condition or procedure that may affect oral study drug absorption * Had major surgery within 3 weeks prior to first dose of study interventions * Clinically significant cardiovascular impairment within 12 months of the first dose of study drug * Active tuberculosis * Has difficulty swallowing capsules or ingesting a suspension orally, or by a feeding tube * Prior treatment with lenvatinib * Prior chemotherapy, targeted small molecule therapy, or radiation therapy within 2 weeks prior to Study Day 1 or has not recovered from adverse events (AEs) due to a previously administered agent. Participants with endocrine-related AEs Grade ≤2 requiring treatment or hormone replacement may be eligible * Has received a live or live attenuated vaccine within 30 days prior to the first dose of study intervention. Note: Administration of killed vaccines is allowed * Previously treated with 4 or more systemic regimens given for recurrent/metastatic disease * Has received an investigational agent or has used an investigational device within 4 weeks prior to study intervention administration * Known psychiatric or substance abuse disorder that would interfere with the participant's ability to cooperate with the requirements of the study