How muddy are we talking about?
Worse than this?
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Not quite as bad like this?
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The only lasting solution is proper drainage. A french drain, or even pea gravel sink holes strategically placed works but it's labour intensive but relatively cheap to do.
Bark mulch is a definite no because of the way it decays; it grows stuff that may not be healthy for the chickens and it won't last long.
If you're going for a cover it up rather than fix, then hardwood chips are standard here in the UK.
The pictures above are of what I inherited when I started looking after these chickens. The ground in the run had years of chicken droppings giving a chicken shite top layer. The rain fell and rather than sinking into the soil, it sat on the surface because the soil was compacted and now coated with a chicken dropping sealant.
At the time, digging in a French drain wasn't an option, nor was a pea gravel sink at the top of the ground slope.
I dug the run. I dug 12 inches to 14 inches deep turning the top layer into the soil below. I still get puddles but they drain reasonably well; leveling the ground properly would stop this. The rest of the ground in the run has stayed acceptable, comparable to the ground outside after a lot of rain on already saturated ground.
If you have space and the trees, collecting leaves in the autumn and dry storing them makes an excellent temporay cover up and also improves the soil once rotted in. I dig the run over a couple of times a year.
I'm in the South West of the UK and we get a lot of rain.