Inclusion Criteria:

Patient must have histologically confirmed squamous cell carcinoma of the head and neck (HNSCC) (excluding SCC of salivary glands, Epstein-Barr virus [EBV]-associated nasopharynx and skin)

Patient must have measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST) version (v)1.1. Measurements must be obtained within 4 weeks prior to randomization

Patient must be >= 18 years of age

Patient must have an Eastern Cooperative Oncology Group (ECOG) performance status 0-1

Patient must have disease progression after prior therapy with an immune checkpoint inhibitor (ICI) in the first-line setting for recurrent/metastatic disease. Patient must have received first-line immune checkpoint inhibition for at least 6 weeks. Patients who have recurred or progressed within 12 weeks of immune checkpoint inhibition administered in the definitive setting for locally advanced disease (for e.g., in the context of a clinical trial) will also be eligible if local therapies are not feasible

Prior combination immunotherapies are permitted, but patient must not have had prior antiangiogenic treatment (e.g., bevacizumab, ziv-aflibercept, ramucirumab, sorafenib, sunitinib, pazopanib, regorafenib, lenvatinib, etc.). Patient must have completed any prior investigational therapy at least 28 days prior to randomization.

NOTE: Patients who received platinum/taxanes in the locally-advanced or recurrent/metastatic setting and did not progress for at least 4 months thereafter, will be eligible for this study. Patients who received cetuximab in the locally-advanced setting and did not progress for at least 4 months thereafter, will also be eligible for this study

Patient must not have a history of >= grade 3 immune-related adverse event on prior ICI therapy (except those that could be managed with steroids [e.g., dermatologic toxicity, asymptomatic elevation of pancreatic enzymes, etc.]) and ICI could eventually be resumed. Patients who developed grade 3 endocrinopathies but are now stable on hormone supplementation and/or a daily prednisone dose of =< 10 mg (or equivalent doses of another glucocorticoid), will be permitted on this trial

Patient must not have a history of PD-1 inhibitor-induced hyper-progression, defined as 100% increase in tumor burden within 8 weeks (or 50% within 4 weeks) of initiating ICI and associated with clinical deterioration

Patient must not have any of the following criteria due to the possibility of increased risk for tumor bleeding with bevacizumab therapy:

Prior carotid bleeding,

Tumors that invade major vessels (e.g., the carotid) as shown unequivocally by imaging studies,

Central (e.g., within 2 cm from the hilum) lung metastases that are cavitary as shown unequivocally by imaging studies,

Any prior history of bleeding related to the current head and neck cancer,

History of gross hemoptysis (bright red blood of 1/2 teaspoon or more per episode of coughing) within 3 months prior to randomization

Patient must not have uncontrolled hypertension, a history of hypertensive crisis or hypertensive encephalopathy, or a history of grade 4 thromboembolism

Patient must not have a history of coagulopathy or hemorrhagic disorders

Patient must not have a history of thrombosis (e.g., pulmonary embolism or deep venous thrombosis) currently requiring therapeutic anticoagulation (prophylactic use of anticoagulation is allowed)

Patient must not be receiving chronic daily treatment with aspirin (> 325 mg/day) or non-steroidal anti-inflammatory agents (NSAID's) known to inhibit platelet function. The use of anti-platelet agents [e.g., dipyridamole (Persantine), ticlopidine (Ticlid), clopidogrel (Plavix)] is allowed only if patient is not receiving concurrent aspirin or NSAID's known to inhibit platelet function.

Patient must have PD-L1 expression >= 1% by combined positive score (CPS) in the tumor and/or immune cells