I was replacing an old ceiling register cover in my house but I found the screw hole on one side of the opening is kind of loose. Is there a way to patch up the mount point so the cover is fully supported?
Break the (non-ignition) end of a balsa-wood kitchen matchstick into the screw hole, then thread the screw into the hole. This will compress the matchstick and provide a snug fit for the screw.
You could probably squirt a bit of caulking or liquid nails in the hole to help tighten it up. Let it dry before putting the register back up.