Epacadostat and Pembrolizumab in Patients With Head and Neck Cancer That Have Failed Prior Immunotherapy

Inclusion Criteria: * Squamous cell carcinoma of the head and neck (which cannot be surgically removed and not amenable to curative intent therapy) * Meet criteria for either the Acquired Resistance OR the Suboptimal Benefit Group \* Acquired Resistance defined as (i and ii must both be met): i. Prior benefit from anti-PD-1/PD-L1 therapy defined as a) prior response, and/or b) ≥5 months of stable disease (SD). Intervening therapies are allowed. ii. Progressive Disease (PD) on recent scans \* Suboptimal Benefit is defined as (i and ii must both be met): i. Prolonged stable disease ≥5 months OR Suboptimal response (\>10% \& \<50% shrinkage per RECIST at any evaluation timepoint) ii. Ongoing stable disease on recent scans iii. Last treatment with an anti-PD-1/PD-L1 agent within 6 weeks prior to starting protocol treatment * Availability of stored tumor tissue OR new tumor biopsy Archival tissue for PD-L1 staining (alternatively a new biopsy (core) at baseline can be used). A minimum of 10 slides is required (unless approval from the PI is obtained) * Measurable disease * Known human papillomavirus (HPV) status * Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 * Be willing and able to provide written informed consent for the trial. * Aged 18 years or older * Demonstrate reasonable organ function * Women of childbearing potential should have a negative urine or serum pregnancy test * Women of childbearing potential should be willing to use 2 methods of birth control or be surgically sterile, or abstain from heterosexual activity for the course of the study through 120 days after the last dose of study medication * Males should agree to use an adequate method of birth control starting with the first dose of study therapy through 120 days after the last dose of study medication Exclusion Criteria: * Currently participating and receiving treatment in a research study or has participated in a study of an investigational therapy and received study therapy or used an investigational device within 2 weeks of the first dose of treatment on this study * Diagnosis of immunodeficiency or is receiving systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to the first dose of trial treatment. * Known history of active TB (Bacillus Tuberculosis) * Hypersensitivity to pembrolizumab, epacadostat or any of its excipients. * Has received prior anti-cancer monoclonal antibody (mAb) within 3 weeks prior to study Day 1, or targeted small molecule therapy within 2 weeks prior to study Day 1, or who has not recovered (i.e., ≤ Grade 1 or at baseline) from adverse events due to agents administered more than 4 weeks earlier. * Has not recovered from prior surgery, chemotherapy or radiation therapy from adverse events due to a previous treatment/ administered agent (i.e., ≤ Grade 1 or return to baseline prior to treatment). Note: Participants with ≤ Grade 2 neuropathy, any grade hearing loss or tinnitus, or typical side effects from radiotherapy are an exception to this criterion and may qualify for the study. Note: If subject received major surgery, they must have recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy. * Has a known additional malignancy that is progressing or requires active treatment. Exceptions include basal cell carcinoma of the skin or squamous cell carcinoma of the skin that has undergone potentially curative therapy or in situ cervical cancer or any tumors that are not likely to influence live expectancy in the subsequent 3 years without active treatment (e.g. low grade prostate cancer in absence of therapy). * Has known active (=growing) central nervous system (CNS) metastases and/or carcinomatous meningitis. Radiation or resected brain metastasis are acceptable if clinically stable. * Has active autoimmune disease that has required systemic treatment in the past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Replacement therapy (eg., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment. * Has known history of, or any evidence of active, non-infectious pneumonitis. * Warfarin use, even if low dose warfarin is not acceptable. However, other anti-coagulants (e.g. aspirin, enoxaparin and heparin derivatives, thrombin inhibitors, etc) are acceptable. * Has a history or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the trial, interfere with the subject's participation for the full duration of the trial, or is not in the best interest of the subject to participate, in the opinion of the treating investigator. * Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial. * Is pregnant or breastfeeding, or expecting to conceive or father children within the projected duration of the trial, starting with the pre-screening or screening visit through 120 days after the last dose of trial treatment. * Has received prior therapy with an IDO inhibiting agent. * Has a known history of Human Immunodeficiency Virus (HIV) (HIV 1/2 antibodies). * Has known active Hepatitis B (e.g., HBsAg reactive) or Hepatitis C (e.g., HCV RNA \[qualitative\] is detected). * Has received a live vaccine within 30 days of planned start of study therapy. * Has received MAO-inhibitors (MAOI) or drug which has significant MAOI activity (meperidine, linezolid, methylene blue) within the 21 days before screening. * History of Serotonin Syndrome after receiving serotonergic drugs. * History or presence of an abnormal electrocardiogram (ECG) that, in the investigator's opinion, is clinically meaningful. Screening QTc interval \> 470 milliseconds is excluded. * Use of any UGT1A9 inhibitor from screening through follow-up period, including the following: diclofenac, imipramine, ketoconazole, mefenamic acid, and probenecid. See Section 5.11 for more details. * History of organ transplant that requires use of immunosuppressives. * Any condition that would jeopardize the safety of the subject or compliance with the Protocol. * Clinically significant cardiac disease, including unstable angina, acute myocardial infarction within 6 months from Day 1 of study drug administration, New York Heart Association Class III or IV congestive heart failure, and arrhythmia requiring therapy