First, I’m curious as to whether your yard has any slope to it. Can the water be drained elsewhere out of the run, or re-directed away from the run?
If you have slope, then see if putting in a diversion trench outside the run to force the flow elsewhere.
If you get blowing rain, then maybe provide a “wall” or barrier of sorts on the side that gets blowing rain to lessen the impact.
If no slope to redirect water, then gravel/ stone as a base will lift your girls outta the mud. My neighbor added gravel to the entirety of his run (15x20 is my guesstimate). He then has been giving them chopped straw on a regular basis for their scratching enjoyment. Of course, a thick layer of varied (non-dyed) mulch would serve similar purpose on top of gravel.