I have a run very similar to yours, shady and at the base of a hill. My run def got mucky, but I found a local tree trimmer who gave me free wood chips. I added about 15 buckets of pine wood chips, it worked wonders. The hens still shove them aside to dig holes for dirt baths etc. But the mud and smell are gone. The poop kinda naturally filters to the base of the wood chips.
Hey i'm sure you can find someone giving chips away for free, its pretty easy to find why not give it a try if it doesn't cost you anything?