How can I repair damaged vinyl flooring over gypsum subfloor?
The white looking stuff is not gypsum, it is the backer that the finish facing is bonded to. If you look closely at it you will see it has the consistency of soft cardboard, it will flake a little and be slightly fibrous.

If you feel you should repair it, make sure the glue is flexible, since the flooring has a little give to it.

I would think if the movers damaged it, they would be responsible to fix it. They should have insurance to cover these items, since they are in the type of business where this type of stuff happens. They also, since that would be the case, have leads on companies that specialize in this type of repair. Take pictures of all the damage and send it to the company that did the moving, they have a reputation they want to keep in good standing, I would hope.