First read this. You are looking at that area and might see something in this that gives you a decent fairly easy fix.
Pat’s Big Ol' Mud Page (fixing muddy runs):
There are two basic concepts. First, stop water from going in. There are a lot of things you can do to help, but when it sets in wet your options her may be limited. It’s just going to get muddy, especially if it is of any size.
The second is get the water out. Evaporation can help a lot once it quits raining but the best way is to let gravity drain it out. That means the water has a place to drain to. You may need to put in a French drain or something like that to take the water to a lower elevation.
Sand drains really well, but not if it is in a bathtub. That’s where a drain comes in really handy. It sounds like the lower half of your run holds water. If you pile enough sand on top of it the water will drain down to that level so it will help. That’s not a permanent fix though. The chickens will scratch that sand everywhere and it will eventually mix in the mud underneath and disappear in the mud. Still it can be a temporary fix. It can be part of a permanent fix too, but you may need to wait until dry weather to do that.
Another temporary fix is to put some type of bedding in there for them to walk on, straw, hay, wood chips, wood shavings, dead leaves maybe. If it stays so wet it gets moldy you may need to remove it and put fresh bedding in, but if it dries out enough to compost, you have a pretty good fix. Many people throw that kind of stuff in the run and let he chickens compost it for them. Different things work for different people.
Another possibility is to put pavers, lumber, pallets, something they can get up out of the mud on. They can spend some time walking in mud and all that but they really need a dry place to go to occasionally. If their feet stay wet they can develop foot problems. Throwing in something like this that you can remove later is a pretty good temporary solution until you can do something more permanent.
Good luck with it. This is a real common problem right now.