Have you read this? It’s pretty good.
Pat’s Big Ol' Mud Page (fixing muddy runs):
https://www.backyardchickens.com/web/viewblog.php?id=1642-fix-a-muddy-run
There are two basic concepts on keeping runs dry. I don’t know what yours looks like so I don’t know what is possible or best for yours. First, try to keep water out to start with. With a decent sized run that can be a real challenge when it sets in wet or during the spring thaw for some of us. Slope cop or run roofs or use gutters and downspouts so water flows away from the run instead of in it, maybe cover the run, or use swales and berms to keep rainwater from flowing in the run. Rain will blow in from the side so maybe a cover on the weather side of a run. Some people use a tarp for that.
The other concept is what you are thinking of, once water gets in, drain it. It has to be high enough for the water to go somewhere, so maybe fill it with a material that drains well to a height that the water has someplace to drain to. Sand drains well but clay is horrible when it comes to draining. Some version of a French drain can help get water away in clay.
There are a couple of problems with pure sand though. Chickens scratch a lot. They will scratch the sand out of the run and into the neighboring area. It will wash out too in a heavy rain. If you fill it with sand you probably want to put some type of barrier around the bottom of the run to contain the sand.
The bigger problem is that the sand will work its way down into the mud underneath and disappear over time. That’s gravity at work. The sand is denser than clay so it will sink. The chickens scratching makes it worse. A way to really slow the sand from sinking is to put a later of gravel down first then put several inches of sand on top of that. It will still disappear over time but a layer of gravel under the sand really makes a big difference. If it has some place to drain to, sand works really well. But don’t make a swimming pool. Don’t just dig out the clay and fill it with sand. It has to be high enough for the water to have some place to drain to.
A lot of runs can be a challenge in the spring especially. Hopefully you can get an idea from this or Pat’s article that can help you.