Will throat or neck cancer show on a neck mri?

Usually. Usually, if the MRI was to look at the neck, and not the cervical spine. Small throat cancers, or cancers without enlarged lymph nodes or masses can be harder to detect. It always helps the radiologist reading and study to have a great clinical history and know what diagnoses the referring clinician and patient are thinking about.
Mri. Yes, will show up on imaging but should also be seen on a good physical exam by an ent.