Drainage is poor in my backyard too, gets soggy & muddy for the chickens. hubby is building a larger coop on a better drained side of the yard. I'll use your idea n get some straw/hay to spread on the new run once it's done (hopefully its completed soon)
I have a few raised garden beds, the veggies do better on the raised beds than the other parts that's just ground.
Do you have any fruit trees?
I too use
raised beds due to the "hard fill" and drainage. A raised bed need not be super high and there are lots of things you can use.
ONE reason I mention this is the recent news stories concerning recycled tires on sports fields and concern about cancer. Personally I would NOT use tires for food plants, only ornamentals. I do have lots of tires dumped on the property too.

I do have one fruit tree and the fruit bushes but they're in another part of the yard. My rhubarb however is in raised beds as it did not do so good where I originally had it. Under the drip line of the my chickens run. I have two varieties and one doesn't seem to do as good as the other. Google "raised garden beds" for lots of ideas of what can be used.
I have an apple tree but so far no apples. I will most likely plant another. Though I have friends who have trees and give me apples, so it's not a high priority.
As for the chicken run I just keep dumping more stuff in as it gets wet and in the summer when it's dry I can harvest the compost. I may "turn" it a few times too. I've read that a certain amount of moisture is needed for compost to "cook".