The purpose of this study is to determine safety and preliminary efficacy of amivantamab monotherapy, amivantamab in addition to pembrolizumab, amivantamab in addition to paclitaxel and amivantamab in addition to pembrolizumab and carboplatin in participants with recurrent/metastatic head and neck cancer. The study will also confirm the recommended Phase 2 combination dose (RP2CD) for amivantamab in addition to paclitaxel. The safety and preliminary efficacy of amivantamab in addition to pembrolizumab will also be determined in perioperative (before and after surgery) setting in participants with resectable locally advanced head and neck squamous cell carcinoma (HNSCC).