Testing Lenvatinib and Cetuximab in Patients With Advanced Head and Neck Squamous Cell Carcinoma and Cutaneous Squamous Cell Carcinoma

Inclusion Criteria: * Histological or cytologic diagnosis of squamous cell cancer * Clinical diagnosis of squamous cell cancer of the head and neck (non-nasopharynx primary tumors: oral cavity, oropharynx, hypopharynx, larynx and sinonasal) or skin * HNSCC and cSCC cannot be curable by surgery and/or radiation therapy * Measureable disease as per RECIST v1.1, which includes locoregional lesions (not amenable to curative surgery and/or radiation) and distant metastatic lesions * Blood pressure \< 150/90 at screening with or without antihypertensive medications and no change in antihypertensive medications within 1 week prior to initiation of treatment * ECOG Performance Status of 0-1 * Adequate renal function as evidenced by calculated creatinine clearance \> 30 ml/min according to the Cockcroft and Gault Formula or by 24 hour urine creatinine clearance * Adequate liver function as determined by (1) Bilirubin \< 1.5 x upper limit of normal (ULN) except for unconjugated hyperbilirubinemia or Gilbert"s syndrome; (2) ALT and AST \< 3 x ULN (\<5 x ULN if subject has liver mets) * Adequate hematologic function as determined by (1) platelets \> 100,000; (2) Hemoglobin \> 9 gm/dl; (3) absolute neutrophil count \> 1200 * Adequate archival tissue to perform molecular analysis through MSK-IMPACT if MSK-IMPACT has not been performed previously on the patient"s tumor; if MSK-IMACT has not been previously performed and adequate archival tissue is not available, a patient should be agreeable to a pre-treatment biopsy Exclusion Criteria: * Prior grade 3 hypersensitivity to cetuximab requiring discontinuation * Prior lenvatinib * Major surgery within 2 weeks of first dose of lenvatinib * Metastatic brain or leptomeningeal tumors (treated metastatic brain or leptomeningeal tumors are allowed). * Anticancer treatment (e.g., radiation therapy, chemotherapy) within 21 days of first dose °An exception is cetuximab treatment, which can be received within 21 days of the first treatment on study * No prior palliative radiation to a target lesion is allowed, unless there is clear biopsy proven progression following radiation. Note, prior radiations to a non-target lesion is allowed. Please see section 9.3.Subjects having a spot Urine Protein:Creatinine ratio of \>1 will undergo 24-hour collection for quantitative assessment of proteinuria. If urine protein \> 1 gram/24 hours, the subject will be ineligible * Significant cardiovascular impairment within 6 months as defined as (1) congestive heart failure greater than New York Heart Association Class II, (2) unstable angina, (3) myocardial infarction; (4) stroke, (5) symptomatic cardiac arrhythmia * On electrocardiogram, QTc interval \> 500 msec * Active infection requiring systemic therapy * Clinically significant hemoptysis or tumor bleeding within 2 weeks prior to first dose of lenvatinib * Other active malignancy except for basal cell or squamous cell carcinoma of the skin, or carcinoma in situ of the cervix or bladder * Women who are breast feeding or pregnant ° Men or women of reproductive potential who are not willing to employ effective birth control from screening to 30 days after the last dose of study drugs; the definition of adequate contraception will be based on the judgment of the principal investigator or a designated associate. For a female patient to be considered as not of child bearing potential, she should fulfill one of the following: ° Post-menopausal women, defined as either women aged more than 50 years and amenorrhoeic for at least 12 months following cessation of all exogenous hormonal treatments, or, women under 50 years old who have been amenorrhoeic for at least 12 months following the cessation of exogenous hormonal treatments, and have serum follicle-stimulating hormone (FSH) and luteinizing hormone (LH) levels in the postmenopausal range for the institution. Or ° Have documentation of irreversible surgical sterilisation by hysterectomy, bilateral oophorectomy or bilateral salpingectomy (but not tubal ligation) * Evidence of clinically significant disease (e.g., cardiovascular, respiratory, gastrointestinal, renal disease) that in the opinion of the investigator could affect the subject safety or interfere with the study assessments