When wet weather sets in it can be hard to keep a run dry. Even if it is covered, rain or snow can blow in from the sides. If you have a small run you may be able to wrap it and keep rain and snow out, but the bigger the run the harder that is.
As others have mentioned drainage is important. If your run is in a low spot where water drains to and stands you have big problem. You may not be able to address that until it dries out this spring or summer. Mine is mostly covered and up on a bit of a rise and drains fairly well but when the weather sets in wet, it can become a muddy mess.
Bedding is supposed to work a lot like a diaper, absorbing the moisture from the poop so it doesn’t stay too wet. If it gets waterlogged it doesn’t work or it might sink down in the mud and disappear. A couple of people have mentioned they sometimes have to replace it.
I agree with Blooie, if straw is what you have straw is what you should use. Try it and see how it works. It may work great or you might wind up topping it off or replacing it.
Something else you can do to get them out of the mud, toss something on top for them to stand on. Cinder blocks or something like that might work though they might soon sink in. Many people use pallets, just put them on top of the mud or maybe on those cinder blocks to help slow them disappearing into the mud. If you have lumber, tree branches, about anything they can stand on to get up out of the mud helps.
They will walk around in the mud and all that, but if their feet stay wet for extended periods of time they can develop feet problems. They need a way to get above the mud at least part of the time.
