First, read this.
Pat’s Big Ol' Mud Page (fixing muddy runs):
https://www.backyardchickens.com/articles/how-to-fix-a-muddy-run-chicken-coop.47807/
There are two basic concepts, keep the water out to start with or get it out once it gets in. Pat's article might give you some ideas for either since you are looking at it and I'm not.
It sounds like your soil may be clay. Clay is pretty much impervious so water will not drain out of it. Chickens often dig holes in it by taking dust baths which can hold water. Clay is a bit challenging to work with.
It also sounds like that area may also be a low spot where water collects. It may not have been that way before the chickens did some scratching. If water collects you can add bedding materials but it will stay wet. As you have seen, adding bedding can be a temporary fix but, at least in your case, it has not been a long term solution.
I don't know the details of that run: how big it is, what materials the soil is, where the water is coming from, or does the water have any place to go. If there is a nearby low spot the fix may be some type of French drain. Dig a trench to that low spot and fill it with sand or gravel. That might make a big difference. I'm guessing your run isn't all that big.
Another option is to build up the run area so it is higher than the surroundings so the water can drain. Try not to create a bathtub that holds water but build it up in a way that it will actually drain and keep water from running into it. My coop is at ground level and the floor is dirt. I added a few inches of clay dirt to the coop floor to stop outside water from running into the coop. It worked, the coop stays dry.
One issue with clay is that rocks or sand will disappear down into the clay over time. You may need to use temporary measures until dry weather to do a permanent fix. Without knowing enough of your details and mainly guessing, my permanent fix would probably involve putting some type of barrier around the bottom of the run to hold your fill in, haul in some more dirt to raise the run floor above the surroundings, add a layer of gravel on top of that to slow the sand from disappearing into the clay, then top that with sand which drains really well. Trying to do this hen it is wet would be a real muddy mess, probably not that successful. I'd suggest pea gravel or river gravel, something that has been worn smooth. If you use crushed gravel the chickens might cut their feet when scratching and get bumblefoot. That's when a splinter or cut gets infected.
Good luck! These things can be a pain.