A Phase II Study of Pembrolizumab in Combination With Circulating Tumor DNA Response-Adaptive Pulsed Chemotherapy in Recurrent/Metastatic Head and Neck Squamous Cell Carcinoma: The SINERGY Trial (Squamous Cell Carcinoma of Head and Neck Response-Guided Therapy)

Inclusion Criteria: * Have clinically confirmed head and neck cancer that is recurrent (comes back/returns to the body) or metastatic (spreads to other parts of the body). * Participants should not have had prior systemic therapy administered in the recurrent or metastatic setting. Systemic therapy which was completed more than 3 months prior to signing consent if given as part of multimodal treatment for locally advanced disease is allowed. * Greater than or equal to 18 years old. * Eastern Cooperative Oncology Group performance status of 0 or 1. * Have measurable disease based on RECIST 1.1 as determined by the site. Tumor lesions situated in a previously irradiated area are considered measurable if progression has been demonstrated in such lesions. * Participants must have normal organ and marrow function as defined by clinical lab values. * Participants must have provided tissue for programmed cell death ligand 1 (PD-L1) biomarker analysis from a core or excisional biopsy (fine needle aspirate is not adequate). Repeat samples may be required if adequate tissue is not provided. A newly obtained biopsy (within 90 days prior to start of study treatment) is strongly preferred, but an archival sample is acceptable. * Measurable disease (either primary site and/or nodal disease) as assessed by Response Evaluation Criteria in Solid Tumors (RECIST 1.1). * Participants must sign a study-specific informed consent form prior to study entry. Participants should have the ability to understand and the willingness to sign a written informed consent document. * Women of childbearing potential must have a negative serum or urine pregnancy test (minimum sensitivity 25 IU/L or equivalent units of HCG) within 24 hours prior to the start of study drug. * Women must not be breastfeeding. * Women of childbearing potential must agree to follow instructions for method(s) of contraception for the duration of treatment. * Men who are sexually active with women of childbearing potential must agree to follow instructions for method(s) of contraception for the duration of treatment with study drug(s). Exclusion Criteria: * Has disease that is suitable for local therapy administered with curative intent. * Has progressive disease (PD) within three (3) months of completion of curatively intended systemic treatment for locoregionally advanced head and neck cancer. * Participants who are receiving any other investigational agents. * Participants in whom signatera ctDNA is not measurable at baseline. * Has had radiation therapy (or other non-systemic therapy) within 2 weeks prior to enrollment or patient has not fully recovered (i.e., ≤Grade 1 or at baseline) from adverse events due to a previously administered treatment. * Note: Participants with ≤Grade 2 neuropathy, ≤Grade 2 alopecia, are an exception to this criterion and may qualify for the study. * Note: If patient received major surgery, they must have recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy. * Active, known, or suspected, autoimmune or inflammatory disorders requiring immunosuppressive therapy, with the exception of low-dose prednisone (\<= 10mg or equivalent). The following are exceptions to these criteria: * Participants with vitiligo or alopecia. * Participants with hypothyroidism (e.g., following Hashimoto syndrome) stable on hormone replacement. * Any chronic skin condition that does not require systemic treatment. * Has a life expectancy of less than 3 months and/or has rapidly progressing disease (e.g. tumor bleeding, uncontrolled tumor pain) in the opinion of the treating investigator. * Participants with a "currently active" second malignancy other than non-melanoma skin cancers. Patients are not considered to have a "currently active" malignancy if they have completed therapy and are free of disease for ≥ 3 years. * Participants with known brain metastases should be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events. * Has had an allogeneic tissue/solid organ transplant. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to pembrolizumab or other agents used in study. * Has received prior therapy with an anti-PD1(anti-programmed cell death protein1) therapy. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements. * Has a history of (non-infectious) pneumonitis that required steroids or current pneumonitis. * Pregnant women are excluded from this study because pembrolizumab is an agent with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with pembrolizumab, breastfeeding should be discontinued if the mother is treated with pembrolizumab. These potential risks may also apply to other agents used in this study. * HIV-positive patients on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions with pembrolizumab, carboplatin, and paclitaxel. In addition, these patients are at increased risk of lethal infections when treated with marrow-suppressive therapy. * Has known active Hepatitis B or Hepatitis C. However, if eradicated participant is eligible. * Has a history of active infection requiring systemic therapy. * Has received a live vaccine within 28 days of planned start of study therapy. Note: Vaccines for COVID-19 are allowed except for any live vaccine that may be developed.