This is the muddy mess I currently have going on. I like to use the smooth indoor-outdoor carpet or the large commercial grade carpet squares with the hard rubber backing on them to cover the bare ground. Both are used material I had on hand. 
I try to stay away from anything abrasive and or sharp to help keep from starting any foot issues. It's not the perfect solution for the muddy mess, but it sure does help and it is easy to rinse all the mud and poo off with a garden hose. I also like to keep extra tubs of clean water so they can flush and keep their little beaks clean.

I try to stay away from anything abrasive and or sharp to help keep from starting any foot issues. It's not the perfect solution for the muddy mess, but it sure does help and it is easy to rinse all the mud and poo off with a garden hose. I also like to keep extra tubs of clean water so they can flush and keep their little beaks clean.

— but a decent enough grade that all water will run off.
If it were me..... I would just haul in more dirt and slope the ground to the edges of your run. Maybe block off small areas of your run and start planting grass? I like to use cheap bird seed in my bare areas of ground and they all love it when it starts grow. 