Inclusion Criteria:
* Diagnosis of histologically or cytologically confirmed recurrent or metastatic HNSCC of the oral cavity, oropharynx, larynx, hypopharynx, or p16+ SCC of the neck (unknown primary).
* Measurable disease per RECIST. Measurable disease defined as lesions that can be accurately measured in at least one dimension (longest diameter to be recorded) as ≥ 10 mm with CT scan, as ≥ 20 mm by chest x-ray, or ≥ 10 mm with calipers by clinical exam.
* Incurable disease, or ineligible for (including patient declined) local therapy.
* At least 18 years of age.
* ECOG performance status ≤ 1
* Life expectancy ≥ 16 weeks.
* Normal bone marrow and organ function as defined below (specimens must be collected within 10 days prior to the start of study treatment):
* Hemoglobin ≥ 10.0 g/dL (criteria must be met without erythropoietin dependency and without packed red blood cell transfusion within the last 2 weeks)
* Absolute neutrophil count ≥ 1,500/mcL
* Platelets ≥ 100,000/mcL
* INR or PT ≤ 1.5 x institutional upper limit of normal (IULN) and aPTT ≤ 1.5 x IULN unless participant is receiving anticoagulant therapy as long as PT or aPTT is within therapeutic range of intended use of anticoagulants
* Total bilirubin ≤ 1.5 x IULN
* AST(SGOT)/ALT(SGPT) ≤ 2.5 x IULN unless liver metastases are present (in which case they must be ≤ 5 x IULN)
* Serum creatinine \<1.5 x ULN or creatinine clearance ≥ 51 mL/min by Cockcroft-Gault or based on a 24-hour urine test
* Known p16 expression, if oropharyngeal primary.
* Postmenopausal or evidence of non-childbearing status for women of childbearing potential: negative urine or serum pregnancy test within 72 hours of Day 1 of study treatment:
\*Postmenopausal is defined as:
* Amenorrheic for 1 year or more following cessation of exogenous hormonal treatments
* Luteinizing hormone (LH) and Follicle stimulating hormone (FSH) levels in the post menopausal range for women under 50
* radiation-induced oophorectomy with last menses \>1 year ago
* chemotherapy-induced menopause with \>1 year interval since last menses
* surgical sterilization (bilateral oophorectomy or hysterectomy)
* Male patients must use a condom during treatment and for 3 months after the last dose of olaparib when having sexual intercourse with a pregnant woman or with a woman of childbearing potential. Female partners of male patients should also use a highly effective form of contraception (see Section 5.6) if they are of childbearing potential
* Ability to understand and willingness to sign an IRB approved written informed consent document (or that of legally authorized representative, if applicable).
Exclusion Criteria:
* Progression within 6 months of curatively intended systemic therapy given for locoregionally advanced disease.
* Investigational therapy within 28 days of treatment start.
* Prior systemic therapy for recurrent or metastatic disease.
* Known CNS metastases/leptomeningeal metastatic disease.
* Other malignancy unless curatively treated with no evidence of disease for ≥ 2 years except: adequately treated non-melanoma skin cancer, curative treated in situ cancer of the cervix, ductal carcinoma in situ (DCIS), stage 1 grade 1 endometrial carcinoma; or low risk-disease per discretion of the PI.
* A history of allergic reactions attributed to compounds of similar chemical or biologic composition to olaparib, pembrolizumab, carboplatin, or other agents used in the study.
* Has received prior therapy with an anti-PD-1, andti-PD-L1, or anti PD-L2 agent or with an agent directed to another stimulatory or co-inhibitory T-cell receptor (e.g., CTLA-4, OX 40, CD137).
* Has received prior radiotherapy within 2 weeks of start of study intervention. Participants must have recovered from all radiation-related toxicities, not require corticosteroids, and not have had radiation pneumonitis. A 1-week washout is permitted for palliative radiation (≤2 weeks of radiotherapy) to non-CNS disease.
* Concomitant use of known strong CYP3A inhibitors (e.g., itraconazole, telithromycin, clarithromycin, protease inhibitors boosted with ritonavir or cobicistat, indinavir, saquinavir, nelfinavir, boceprevir, telaprevir) or moderate CYP3A inhibitors (e.g., ciprofloxacin, erythromycin, diltiazem, fluconazole, verapamil). The required washout period prior to starting olaparib is 2 weeks.
* Concomitant use of known strong CYP3A inducers (e.g. phenobarbital, enzalutamide, phenytoin, rifampicin, rifabutin, rifapentine, carbamazepine, nevirapine, and St. John's wort) or moderate CYP3A inducers (e.g., bosentan, efavirenz, modafinil). The required washout period prior to starting olaparib is 5 weeks for enzalutamide or phenobarbital and 3 weeks for other agents.
* Resting ECG indicating uncontrolled, potentially reversible cardiac conditions as judged by the investigator (e.g., unstable ischemia, uncontrolled symptomatic arrhythmia, congestive heart failure, QTcF prolongation \> 500 ms, electrolyte disturbances, etc.) or patients with congenital long QT syndrome.
* Diagnosis of myelodysplastic syndrome/acute myeloid leukemia or with features suggestive of MDS/AML.
* Major surgery within 2 weeks of starting study treatment; must have recovered from any effects of major surgery.
* Has a diagnosis of immunodeficiency or is receiving chronic systemic corticosteroid therapy (in doses exceeding 10 mg daily of prednisone equivalent) or any other form of immunosuppressive therapy within 7 days prior to the first dose of treatment.
* Received a live vaccine or live-attenuated vaccine within 30 days prior to the first dose of pembrolizumab. Administration of killed vaccines is allowed. Examples of live vaccines is allowed. Seasonal influenza vaccines for injection are generally killed virus vaccines and are allowed; however, intranasal influenza vaccines (e.g. FluMist®) are live attenuated vaccines and are not allowed.
* Considered a poor medical risk due to a serious, uncontrolled medical disorder, non-malignant systemic disease, or active, uncontrolled infection. Examples include, but are not limited to, uncontrolled ventricular arrhythmia, recent (within 3 months) myocardial infarction, uncontrolled major seizure disorder, unstable spinal cord compression, superior vena cava syndrome, extensive interstitial bilateral lung disease on high resolution computed tomography (HRCT) scan or any psychiatric disorder that prohibits obtaining informed consent.
* Unable to swallow orally administered medication or with gastrointestinal disorder likely to interfere with absorption of study medication.
* Pregnant and/or breastfeeding. A WOCBP who has a positive urine pregnancy test within 72 hours of first dose of treatment. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required.
Female patients who are expecting to conceive or Mmale patients who are expecting to father children within the projected duration of the study, starting with the screening visit through 120 days after the last dose of trial treatment.
* Prior organ or allogeneic stem cell transplant or double umbilical cord blood transplantation.
* Has an active autoimmune disease (i.e. rheumatoid arthritis, lupus, Sjogren's syndrome) that has required IV or subcutaneous systemic treatment in the past 2 years (excluding Rituxan). Replacement therapy (i.e. thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment.
* Has a known history of Hepatitis B (defined as Hepatitis B surface antigen \[HBsAg\] reactive) or known active Hepatitis C virus (defined as HCV RNA \[qualitative\] is detected) infection. Note: no testing for Hepatitis B and Hepatitis C is required unless mandated by local health authority.
* History of (non-infectious) pneumonitis/interstitial lung disease that required steroids or has current pneumonitis/interstitial lung disease.
* Has a known history of active TB (Bacillus Tuberculosis).
