First read this article.
Pat’s Big Ol' Mud Page (fixing muddy runs):
https://www.backyardchickens.com/web/viewblog.php?id=1642-fix-a-muddy-run
With the coop and run in the low spot where water drains to instead of away from, you have a challenge in front of you. There are two general ways to attack this problem.
Try to keep as much water out as possible. Maybe roof the run and slope roofs or use gutters to get the water away. That probably won’t help you a lot. Can you use berms or swales to divert rainwater away from that area? Building the area up higher so water doesn’t run in can also work. I installed a swale and put a few inches of clay dirt in my coop to build it up to keep water out. I don’t know how big your run is. Concrete sounds a bit dramatic, but it could work.
Drain the water out once it gets in. I don’t know what your “soak” would look like but I assume it is what I’d call a French drain. If you have an area to drain the water to, that would work great. It could help with the muddy area around the run also. Working with berms and swales to divert rainwater runoff and using a soak are probably your best long term solutions, not sure though because I am not familiar with the lay of your land.
Filling the run with sand is a great idea, but there are a few complications. Sand will drain really great as long as it is high enough to have a place to drain to but over time it will work is way down into the clay underneath so you will need to add more sand as required. The chickens will scratch in it, scratching some outside the run which leads to adding sand. Sand can wash out if you get a heavy rain, so it can wind up level with your surroundings, not raised above so it can drain. A way to help with that is to put a barrier around the bottom of your run to contain the sand to keep it from washing out and reduce how much they scratch out. Pat talks about how you can put a layer of gravel down first to reduce how fast the sand works its way into the clay.
Good luck with this. You can really have success with this problem but with a really big run it can really be challenging.