This phase II trial studies the effect of re-irradiation with NBTXR3 in combination with pembrolizumab in treating patients with head and neck squamous cell cancer that cannot be removed by surgery (inoperable) and has come back (recurrent). NBTXR3 is a drug that is designed to improve the effectiveness (how well something works) of radiation therapy. The drug is injected into a tumor and activated (turned on) by radiation. Stereotactic body radiation therapy uses special equipment to position a patient and deliver radiation to tumors with high precision. This method may kill tumor cells with fewer doses over a shorter period and cause less damage to normal tissue. Radiation therapy, such as intensity modulated radiation therapy or intensity modulated proton therapy, uses high energy to kill tumor cells and shrink tumors. Immunotherapy with monoclonal antibodies, such as pembrolizumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Giving NBTXR3 activated by radiation together with pembrolizumab may help to control head and neck squamous cell cancer.

PRIMARY OBJECTIVES: I. To estimate progression free survival (PFS) and early clinical benefit in patients treated with hafnium oxide-containing nanoparticles NBTXR3 (NBTXR3) activated by stereotactic body radiation therapy (SBRT) reirradiation, with concurrent pembrolizumab. II. To assess the safety profile and estimate early clinical benefit of NBXTR3 activated by dose reduction intensity modulated radiation therapy (IMRT) or intensity modulated proton therapy (IMPT) reirradiation with concurrent pembrolizumab, in subjects with locoregional recurrent head and neck squamous cell carcinoma (HNSCC) not eligible for SBRT. SECONDARY OBJECTIVES: I. To evaluate tumor response after NBTXR3 activated by SBRT reirradiation with concurrent pembrolizumab. II. To evaluate tumor response after NBTXR3 activated by dose reduction IMRT/IMPT reirradiation with concurrent pembrolizumab. III. To evaluate the safety profile of NBTXR3 activated by SBRT reirradiation with concurrent pembrolizumab. IV. To evaluate time-to-event outcomes of NBTXR3 activated by SBRT reirradiation with concurrent pembrolizumab. V. To evaluate time-to-event outcomes of NBTXR3 activated by dose reduction IMRT/IMPT reirradiation with concurrent pembrolizumab. EXPLORATORY OBJECTIVES: I. To evaluate lymphedema/fibrosis \& dysphagia-related toxicities and functional outcomes of treatment with NBTXR3 activated by SBRT or IMRT or IMPT reirradiation and concurrent pembrolizumab. II. To assess functional and patient reported outcomes (PRO) of treatment with NBTXR3 activated by SBRT or IMRT or IMPT reirradiation and concurrent pembrolizumab. III. To associate radiomic measurements with outcomes of treatment with NBTXR3 activated by SBRT or IMRT or IMPT reirradiation and concurrent pembrolizumab. IV. To evaluate biomarkers of response in subjects treated with NBTXR3 activated by SBRT or IMRT or IMPT reirradiation and concurrent pembrolizumab. OUTLINE: Patients are assigned to 1 of 2 cohorts. COHORT I: Patients receive NBTXR3 intratumorally (IT) on day 1. Patients then undergo SBRT every other day (QOD) on days 15-29. Beginning the first day of radiation therapy, patients also receive pembrolizumab intravenously (IV) over 30 minutes on day 1. Cycles repeat every 3 weeks for up to 2 years in the absence of disease progression or unacceptable toxicity. COHORT II: Patients receive NBTXR3 IT on day 1. Patients then undergo IMRT/IMPT every day (QD) on days 15-50. Beginning the first day of radiation therapy, patients also receive pembrolizumab IV over 30 minutes on day 1. Cycles repeat every 3 weeks for up to 2 years in the absence of disease progression or unacceptable toxicity. After completion of study treatment, patients are followed up every 3 months for 2 years and then every 6 months for up to 5 years.