Hi again, sounds like you have an old style of breeze block, try filling the hole you already have with a liquid nails type product (GRIPFIL is very good), then push the wall plug into it and let it set, when its dry fix the screw into it in the normal way.
I would let it dry for a couple of days, not just touch dry.
If you are just trying to hang a picture I would try masonary nails, these are much stronger than normal nails and will enter a breeze block with ease.
Your drill bit should have a tip soldered onto the front of it (this is a masonary bit) standard wood or metal cutting drills will not do the job. Put your drill onto the hammer setting, this helps drive the drill bit into the wall.
Another tip is to put a small drill bit in first to give your larger drill bit a guide to follow.
Best of luck.
johnny edge