This study is designed to learn more about the effects of Petosemtamab, an investigational drug not FDA approved, compared to standard of care treatment for people with head and neck cancer that has been previously treated with immunotherapy and chemotherapy. If decided to take part in this study, participants will go through a screening period, treatment period, and follow-up period. During the screening period following signing of consent form participants will be evaluated for screening criteria and determined if they qualify for the study. During treatment period participants will be randomly assigned to either receiving petosemtamab or one of the three standard of care choices. In the follow up period the side effects after completion of petosemtamab administration or assigned standard of care treatment will be evaluated. Also, approximately every 12 weeks after study treatment stopped, either petosemtamab or assigned standard of care, participants will receive a phone call for long term follow-up for up to 1.5 years. Participation in this study may last up to two years. The participants will complete questionnaires to evaluate the differences in the quality of life for participants that received petosemtamab versus standard of care treatment. Some serious risk related to this study are infusion related reactions, rash and diarrhea. There may not a benefit from joining the study. The head and neck cancer may improve while on this study but it may not, and it may even get worse. It will also help inform how well Petosemtamab works at curing this type of cancer. The study results may be used to help others in the future.

The main purpose of this study is to measure overall survival (OS) and safety of ivonescimab (study drug) when combined with chemotherapy drugs carboplatin, paclitaxel or nab-paclitaxel compared to pembrolizumab combined with chemotherapy drugs carboplatin, paclitaxel or nab-paclitaxel. Participants will undergo screening procedures done to determine if they meet the requirements to be in this study. Screening will be completed within 28 days before receiving the study drug. Many of these screening measures are likely part of regular cancer care and may be done even if it turns out that you do not participate in the research study.

Once enrolled in the study, participants will visit the clinic every three weeks for 4 cycles of ivonescimab plus chemotherapy or pembrolizumab and chemotherapy for up to four infusions, followed by ivonescimab or pembrolizumab every three weeks for up to 24 months. If a participant's physician decides to use nab-paclitaxel chemotherapy for the first 4 cycles of treatment, the schedule of treatment is different and will require that the participant comes to the clinic for this infusion on days 1, 8, and 15 of each cycle. There will be follow-up check-up visits with the study team approximately 7 days, 30 days and 90 days after the last treatment or before the participant starts a new treatment for the cancer. Ninety (90) days after the participant stops taking the study drug, there will be a call or a visit scheduled to review how the they are feeling. This is a survival call/visit and will happen every 90 days until the end of the study. Participation in this study will last about 4 years, 2 years in active treatment and 2 years in follow up.