iPhone 5c is not found by iTunes
The "setup as new" or "restore from backup" choice in iTunes is a bit of a misnomer. Setup this phone as new sounds like it's going to do something like wipe the phone, but it really just means the phone hasn't been plugged in and used with iTunes, so add it as a new phone into iTunes.

You definitely do not want to restore from backup (overwriting what you have on the phone now), so choose setup as new.

A couple of references, since I know "setup as new" sounds scary:


https://discussions.apple.com/thread/4094634
https://deciphertools.com/support/knowledge-base/how-do-i-backup-my-iphone-or-ipad-w-itunes/ (from my blog - very bottom of the article)