Is there a spot you can dig a small trench on the outside of the run that will let the water out before you get mud in the run? I get it, my run is also covered but still get sideways rain and water that comes up from the ground in the wet season.
If your run is getting mud wet adding material to it will help but the original problem will still exist. Adding more and more stuff is only going to add up to a thick layer of yuck.
My advice for a muddy run is as always Ditched, Dams, Drains, Gutters and Roofs.