Could a pinched nerve in my neck cause facial numbness?

Unlikely: Sensation in the face is mediated through cranial nerves which are separate from spinal nerves. Depending on the type of pain that you have, one of the facial neuralgia disorders might be associated.

Not likely: Not likely. The nerve that supplies sensation to the face is the trigeminal nerve. This is a cranial nerve that exits the skull base behind the eyes, so not related to any cervical nerve rootlets. It would be entirely abnormal to have numbness over the face from a pinched nerve in the neck.