What can I do about my deteriorating basement floor with a large hole that seems to go right through to clay?
Well, lacking a picture for now (try it again, please, and don't delete the link text it puts in on the bottom - that won't show up in your post, but it will let us see the picture) If you have a concrete floor now, you can break out bad concrete until you get to good concrete, put in some crushed rock for a sub-base and pour new concrete. Depending what's going on, you may want to break out some more floor and install drain pipes and a sump pit before you backfill and concrete over it. Avoid getting too close to walls, chimneys or columns/posts when breaking things out, or call in an engineer if it seems like you need to.

OK - the lower bit of concrete with the tidy corner is presumably a footing - so don't mess with that. The floor, you can basically do as described above - cut back to a sound edge and fill in with fresh concrete, or fiddle with drainage as seems appropriate. Many older homes were built with dirt -floored basements, and a concrete floor was an afterthought, often poorly done on the cheap by some later homeowner. If you just want to plug the hole, see above. If you have higher aspirations for your basement, describe them and we'll give it a thought.