Chicken poop and clay are both so crusty when they dry. What a mess for you! It's hard to deal with when you're in the middle of the rainy season. When things get better, I'd add a thick layer of sand on top, to cover the dirt that turns into mud and provide more drainage for the surface. You'll probably need to add a board or something to retain the sand, so it doesn't wash out. Sand can also be raked, to remove excess chicken poop and feathers.
Have you thought about tarping at least part of the run? Then at least part of the run would be drier for them. Pallets would at least get them out of the mud part of the time. You could try dumping something like a heavy layer of wood chips or straw in the run for now. If you've got so much water that it's got puddles, you might end up with a pretty gunky mess with that, too.