Inclusion Criteria:  Participants must meet the following criteria on screening examination to be eligible to participate in the study: Participants must have histologically confirmed squamous cell carcinoma of the head and neck (SCCHN) with evidence of recurrent, metastatic (R/M) or advanced, incurable disease from any mucosal subsite including oral cavity, oropharynx, larynx, hypopharynx, nasal cavity, and the paranasal sinuses. Participants must have at least one RECIST v1.1 measurable lesion, as defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) ≥1 cm with CT scans or MR imaging.  Must have had at least 1, but no more than 2, prior lines of prior systemic therapy for R/M SCCHN; one of these lines should have included PD-1/L1 blockade  Platinum-based therapy as part of definitive/adjuvant or curative-intent treatment can count as 1 prior line of therapy if the subject progressed within 6 months of receiving therapy. At least 2 weeks must have elapsed since the end of prior chemotherapy, biological agents (3 weeks for anti-cancer monoclonal antibody containing regimens) or any investigational drug product, with adequate recovery of treatment-related toxicity to NCI CTCAE Version 5.0 grade ≤1 (or tolerable grade 2) or back to baseline (except for alopecia or peripheral neuropathy). Be ≥18 years of age on the day of signing informed consent. Must provide prior data on tumor PD-L1 expression status and HPV status, if available Have a performance status of 0 or 1 on the ECOG Performance Scale  Participants must have adequate organ and marrow function as defined below (within 14 days prior to study registration):  absolute neutrophil count ≥ 1,000/mcL hemoglobin ≥ 9 g/dL platelets ≥ 100,000/mcL total bilirubin ≤ upper limit of normal (ULN) AST(SGOT)/ALT(SGPT) ≤ 2.5x institutional ULN (or ≤ 1.5x institutional ULN if concomitant with alkaline phosphatase >2.5x institutional ULN) or ≤ 5x ULN for those with liver metastases serum creatinine ≤ 1.5x ULN OR creatinine clearance ≥ 60 mL/min/1.73 m2 for participants with creatinine levels above 1.5x ULN coagulation profile INR ≤ 1.5x ULN unless the participant is receiving an anticoagulant Baseline tumor measurements must be documented from imaging within 28 days prior to study registration. Female subjects of childbearing potential should have a negative urine or serum pregnancy test within 7 days of study registration. Female subjects of childbearing potential should have a negative urine or serum pregnancy test repeated within 72 hours prior to receiving the first dose of study medication. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required. Female and male subjects of childbearing potential must agree to use an adequate method of contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and 4 months after completion of stud drug administration. Contraception is required before starting the first dose of study medication through 120 days after the last dose of study medication. Note: Abstinence is acceptable if this is the usual lifestyle and preferred contraception for the subject. Be willing and able to provide written informed consent for the trial.  Exclusion Criteria:  Participants who exhibit any of the following conditions at screening will not be eligible for admission into the study.  Have been previously treated with 3 or more lines of systemic therapy for R/M SCCHN.  -- Have received treatment with a prior PI3K pathway inhibitor  Have received radiation therapy (RT) within 14 days of the first dose of duvelisib on study. Participant has known active central nervous system (CNS) metastases and/or carcinomatous meningitis. Subjects with previously treated brain metastases may participate provided they are stable (without evidence of progression by imaging and off systemic steroids for at least 4 weeks prior to the first dose of study treatment), and have no evidence of new or enlarging brain metastases. This exception does not include carcinomatous meningitis which is excluded regardless of clinical stability, because of the poor prognosis and progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events. Concurrent administration of other cancer specific therapy or investigational agents during the course of this study. Uncontrolled intercurrent illness including but not limited to ongoing or active infection; evidence of symptomatic congestive heart failure, unstable angina pectoris, stroke, or ventricular arrhythmia within 6 months of enrollment. Have received a live or live attenuated vaccine within 4 weeks of the first dose of duvelisib. Have received medications or consumed foods that are strong inhibitors or inducers of cytochrome P450 (CYP3A) within 2 weeks of, or while on, duvelisib.