Combined Amivantamab, Carboplatin and Paclitaxel in Unresectable Locally Recurrent or Metastatic Head and Neck Cancer

Inclusion Criteria: * Participants must have histologically documented squamous cell head and neck carcinoma that meets one of the following criteria; was metastatic at diagnosis, persisted, metastasized or recurred following definitive treatment, or recurred locally and participant has either a contraindication to surgical treatment of lesions (i.e., complete resection is not possible or not expected to be clinically beneficial or resection confers significant functional concerns) or has refused surgical or radiation treatment. Both human papillomavirus (HPV) positive and negative are eligible. Participants with cancers originating from the wet lip, oral cavity, oropharynx, larynx, hypopharynx, nasopharynx, epiglottis, and nasal cavity/paranasal sinuses are eligible. * Participants must have measurable disease documented by CT or MRI. Measurable disease must be assessed within 28 days prior to registration and non-measurable disease must be assessed within 42 days prior to registration. Pleural effusions, ascites and laboratory parameters are not acceptable as the only evidence of disease. Participants whose only measurable disease is within a previous radiation therapy port must demonstrate clearly progressive disease (in the opinion of the treating investigator) prior to registration to be considered measurable. * Participants must have received prior therapy with a Programmed death ligand 1 (PD-(L)1) inhibitor or PD-1 inhibitor for treatment of head and neck cancer (in any setting). * Participants must be ≥ 18 years old at the time of registration. * Participants must have ECOG/Zubrod Performance Status of 0-1. * Participants must have a complete medical history and physical exam within 28 days prior to registration. * Participants must have adequate organ and marrow function within 28 days prior to registration: * Participants must have a calculated creatinine clearance ≥ 45 mL/min using the Cockcroft-Gault Formula. The specimen for this test must have been drawn and processed within 28 days prior to registration. For creatinine clearance formula see the tools on the CRA Workbench https://txwb.crab.org/TXWB/Tools.aspx. * Participants with known human immunodeficiency virus (HIV)-infection must be on effective anti-retroviral therapy and have undetectable viral load test on the most recent test results obtained within 6 months prior to registration. An undetectable viral load is defined as (1) CD4+ cell count greater or equal to 200 cells/mm3; (2) No history of non-malignancy AIDS defining conditions other than historical low CD4+ cell count. * Participants with a history of hepatitis C virus (HCV) infection must have been treated and cured. Participants must have an undetectable HCV viral load test on the most recent test results obtained within 6 months prior to registration, if indicated. * Participants must be offered the opportunity to participate in specimen banking. With participant consent, specimens must be collected and submitted via the SWOG Specimen Tracking System. * Participants or their legally authorized representative must be informed of the investigational nature of this study and must sign and give informed consent in accordance with institutional and federal guidelines. For participants with impaired decision-making capabilities, legally authorized representatives may sign and give informed consent on behalf of study participants in accordance with applicable federal, local, and WCG IRB regulations. Exclusion Criteria: * Participants must not have received more than two prior systemic therapy regimens for recurrent and/or metastatic disease. NOTE: Systemic treatment administered with radiotherapy for radiosensitization in the palliative setting will not be considered a line of therapy. * Participants must not have had prior anti-cancer therapy within 14 days of registration. Note: Participants must not be planning to receive any concurrent chemotherapy, immunotherapy, biologic, or hormonal therapy for cancer treatment while receiving treatment on this study. * Participants must not have received treatment with a platinum or taxane agent as part of a prior treatment for recurrent or metastatic disease. NOTE: Platinum or taxane agents administered with radiotherapy for radiosensitization in the curative or palliative setting is allowed if there was no evidence of progression within 6 months of last dose of platinum or taxane. * Participants must not have had prior treatment with cetuximab or another Epidermal Growth Factor Receptor (EGFR) inhibitor in the locally recurrent advanced unresectable or metastatic setting. * Participants must not have known hypersensitivity and/or contraindication to carboplatin or paclitaxel in the opinion of the treating investigator. * Participants must not have active cardiovascular disease including but not limited to: Participants with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, must have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, participants must be class 2 or better. Participants must not have experienced arterial thromboembolic events, including but not limited to myocardial infarction, transient ischemic attack, cerebrovascular accident, or unstable angina, within 6 months prior to registration.Participants must not have evidence of preexisting uncontrolled hypertension 28 days prior to Step 1 registration as documented by baseline blood pressure reading with systolic blood pressure \>160 mmHg and/or diastolic blood pressure \>100 mmHg. Participants on antihypertensive therapies with controlled blood pressure are eligible. Participants must not have congestive heart failure (CHF), defined as New York Heart Association (NYHA) class III-IV or hospitalization for CHF (any NYHA class) within 6 months prior to registration. * Participants must not have an uncontrolled illness, including but not limited to: Participants must not ongoing uncontrolled diabetes as determined by the treating investigator prior to registration. Participants must not have ongoing or active infection or be diagnosed with or suspected of having a viral infection as determined by the treating investigator. NOTE: Participants who have an infection requiring antimicrobial therapy will be required to complete antibiotics 7 days prior to starting treatment. Participants must not have impaired oxygenation requiring continuous oxygen supplementation as determined by the treating investigator. Participants must not have a history of pneumonitis that required drug therapy or active symptomatic ILD/pneumonitis, including drug-induced or radiation ILD/pneumonitis. Participants must not have any ophthalmologic condition that is clinically unstable in opinion of treating investigator. Participants must not have active bleeding diathesis as determined by the treating investigator prior to registration. Participants must not have any psychiatric illness, social situation, or any other circumstances that would limit compliance with study requirements in opinion of treating investigator. * Participants must not have had major surgery (excluding placement of vascular access, percutaneous feeding tube replacement, tracheostomy, or tumor biopsy) or had significant traumatic injury within 28 days prior to registration. All adverse events associated with prior surgery must have resolved to ≤ Grade 1 (per CTCAE 5.0) prior to registration. NOTE: Participants must not have surgery planned during the time the participant is expected to participate in the study. Planned surgical procedures to be conducted under local anesthesia are allowable. * Participants with a known history of chronic hepatitis B virus (HBV) infection must have undetectable HBV viral load while on suppressive therapy on the most recent test results obtained within 6 months prior to registration, if indicated. * Participants must not have other clinically active infectious liver diseases. * Participants must not have a concurrent malignancy whose natural history or treatment (in the opinion of the treating investigator) has the potential to interfere with the safety or efficacy assessment of the investigational regimen. * Participants must not be pregnant or nursing (nursing includes breast milk fed to an infant by any means, including from the breast, milk expressed by hand, or pumped). Individuals who are of reproductive potential must have agreed to use an effective contraceptive method with details provided as a part of the consent process. A person who has had menses at any time in the preceding 12 consecutive months or who has semen likely to contain sperm is considered to be of "reproductive potential." In addition to routine contraceptive methods, "effective contraception" also includes refraining from sexual activity that might result in pregnancy and surgery intended to prevent pregnancy (or with a side-effect of pregnancy prevention) including hysterectomy, bilateral oophorectomy, bilateral tubal ligation/occlusion, and vasectomy with testing showing no sperm in the semen.