GitHub does have some API endpoints to create, update, or get references: https://developer.github.com/v3/git/refs/

These endpoints can be useful for users to recreate some git "plumbing" functionalities over the Gitea API. With these endpoints, users could reset branches for example.

More generally, GitHub has a "Git Database API" which works at a lower level and allows user to do almost anything, but that might be a bigger project: https://developer.github.com/v3/git/