If you really want a permanent fix to a muddy run, It will take some doing, but a sand over gravel cover run that is properly drained is difficult to beat.
You need to dig all of that mud out of the run to whatever depth it takes to get rid of the mud and find sound soil. Then dig in a french drain that fits a 4" perforated PVC drain pipe, and run that ditch to wherever you need to for the water to drain. That may involve a drain field or even a gravel cistern depending on your layout. Cover the pipe with a good quality of landscape cloth to protect it, an bury the pipe trench in 3/4" gravel, then lay in at least a 4" bed of more gravel over the top of that. Over the gravel, put as much sand as you need to fill the run. My 50 square foot run took a yard of gravel and a bit over a yard of coarse sand to top it off. (See my foundation page for details and pictures)
Now I have a
covered run, so water drainage is not a problem. I put a fair amount of Feed Grade DE in the sand, and between the two, chicken poop is completely dessicated, and ground into nothing. I go out and try to scoop it periodically, but there are no chunks left after a day or so. The chooks help break everything down with their scratching, and there is no mess and no mud whatsoever in the run. Granted I have a small flock and the covered run was an economical and easy to build solution; the fact that it is perfectly dry all of the time is the main factor in why it does not stink and why it stays so clean. It's that old time vs money equation and it depends on what you and your chook's needs are, but I am convinced that a sand run is about the only way to go, especially in the backyard...
It makes for an excellent Spring project, and if you call around you might be amazed at how cheap gravel and sand are delivered....