Re-drilling into existing hole in plasterboard/breeze block
Another option is to use a pair of holes. I often do this for larger pictures and mirrors, it spreads the load over two fixings and helps stabilisation, no tipping out of plumb or level when the door slams or such like.
Re. the original hole, you're going to struggle to fill it well enough to be able to drill half of it out again, especially if it's more than 50mm deep. Difficult to picture half a hole making such a difference without seeing the location.