Read this. It’s got some good ideas.
Pat’s Big Ol' Mud Page (fixing muddy runs):
https://www.backyardchickens.com/web/viewblog.php?id=1642-fix-a-muddy-run
It’s not good but when it sets in rainy or during the thaw it’s something you might have to live with a bit. However, they do need to be able to stand on something dry. They can get blisters on the bottom of the feet if they live in wet. So put something out there that they can occasionally stand on that is up out of that mess.
I don’t know what your set-up is but there are two basic concepts. I know it’s too late for a lot of these but maybe one or two will give you an idea.
The first is to try to stop water from getting in there to start with. Slope coop and run roofs so they don’t drain in there or use gutters to get that water away. Use berms or swales to divert water from running in there. Maybe a roof or some kind of rain guard on the sides. A lot of rain will blow in from the side.
The other is to give water that gets in a place to go. Don’t put your coop or run in a low spot where water drains to or stands. Build the run level up so it is higher than the surroundings. I added a few inches of clay dirt to my coop floor to make sure it was higher than the surroundings. But in a run where rainwater will probably blow in sand will be better since it drains so much better. Part of the problem is that clay will hold water while sand will help it drain out. The clay in my coop will stop water from soaking in but sand will help it soak out of a run. As long as the water has a place to drain to, a French drain may be real beneficial. What you can do depends a lot on your terrain.
I’ve got a roof over my run, put it where it is a bit high so it can drain, and built a swale to keep rainwater runoff out, but it’s still a muddy mess when it rains a bunch. Part of it is high enough and under that roof to at least give them a dry place to go and I dumped some pea gravel at the door from the coop so I have a more solid place to walk. But they will still dig holes for dust bathing which become mud puddles.
Good luck with it. Pat’s article might give you some ideas.