I know I'm super late to the party, but I'd still like to contribute more ideas...
Like a french drain? I'm not sure the exact procedure, but I bet it would help.
I think a very overlooked factor is the additional rainwater runoff you'll get from the coop roof.
You definitely need a gutter. If the ground already has a tendency to get saturated, all that runoff will pool on the ground and seep (or flood) right into the run area. It has to be directed somewhere else. I have a rain chain and collection bucket with a hose that leads to my garden beds. Before that was set up, I just had a temporary 5 gallon bucket to hold the gutter runoff, and during heavy rain spells it was overflowing in just a few hours!
That is exactly what my chicken yard used to look like before it became the chicken yard! It was lined with landscape fabric and topped with bark mulch when we bought our house. I put my birds there for a short time, but they scratched down to the fabric and turned it into giant fluffy remnants floating all over and left the dirt exposed, which turned to mud when it rained. Needed a better solution.
Like others have said, chunky material like big wood chips are the way to go. It'll let the water drain through and they hold up well. But having used wood chips in my horse paddock in the past and watching them just sink away and disappear over a few short months, I wasn't sure it would work with chickens any better long term, because as you know, chickens are excellent mixer-uppers of dirt and bedding. It's really rainy here in Oregon and I wanted clean wood chips, not muddy.
Important to mention: This was for the exposed yard area, not my covered run.
So I copied this
mud management idea from Newland Poultry in the UK.
What makes this system work so well is that the ground (mud) and the surface material (wood chips) is kept separate by a sturdy barrier... I used PVC poultry netting. First I removed all the old landscape fabric and mulch (a terrible chore), then laid the netting, attached it to the ground using garden pins, and put wood chips on top.
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The wood chips never get mixed with the muddy ground below, and it's not slippery or mucky or stinky at all! It's amazing! My chickens still enjoy scratching and finding bugs, and the PVC material won't hurt their feet or turn into floaty fluff balls. These chips should last a couple/few years before they start to break down enough to be replaced. Then I'll reuse the old as mulch for my other landscaping.
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@MandaRae you have so much beautiful grass, it seems a shame to turn it into wood chips. But chickens do have a way of turning grass into dirt pretty quick. I think you should definitely plan some gutters and a system to divert the water away.