I drilled holes for drywall anchors too wide - so now the anchor and the screw is loose. How do I fix this?
This is very complicated, and I'm hoping that you'll stick through all of the steps with me. ;)
(No, really, that's the answer. You can't patch the drywall and put anchors in; the most you could hope for would be re-drilling holes near to your anchor, but you really risk cracking the drywall if you do that. I'd just get bigger anchors.)
You could switch to a toggle anchor. It requires a larger hole to install because the wings need to pass through, but it will hold much better than a plastic anchor.
They won't work if you need the screw head to protrude from the wall (to hang a picture wire, for example) because you screw it all the way down so the wings are pressed against the inside of the wall.
That said, If I need to hang a fixture, and I can use a toggle instead of the plastic anchors that came with it, I'll discard the plastic anchors and use a toggle anchor.