I know what you mean about a mucky run. I've got 6 hens and a run (not covered) about the size of a single garage. Compacted, a bit smelly and not much fun to get eggs from.
I've just started copying a friend who has no muck, beautiful soils and very happy chooks.
What my friend does is lime her run regularly, which keeps it from smelling and changes the pH to something more neutral. She also put a really thick layer of pea straw or hay - 25 cm or so - so the chooks always have to scratch and look for any grain, pellets, scraps they are fed. Somehow the combination of lots of loose mulch cover and lime makes this amazing soft black soil. I've been scratching my head about why this would work so well. Might also have something to do with the fact that she is adding a lots of carbon (ie as straw) to her run, which means she's hitting the classic ideal carbon:nitrogen ratio that is promoted for ideal compost heaps. However it works, it does, and she runs twice as many hens in the same size space I have!
I tried bark mulch last winter in my run, but it looked OK for 10 days then just got compacted. I now have pea gravel at the gate and next to the egg box, as most of the compaction there comes from people not chooks.
Just two days ago I gathered all the light humus-making material I could find - some straw, several wheelbarrow loads of autumn leaves, dried grass, and chucked it in. Happy hens scratching about now but might be the critters I found with the leaves. Also I scattered about 15 handfuls of dolomite lime. Will try to track down some more fluffy cover like pea straw (but I'm stingy...).
I'm keen to crack it as my friend harvests great soil as well as eggs and I want to do the same.
Regards, Pip