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I would use the area between garden and play area. No matter where you put it, you’re going to have to maintain their space. It’s quite easy really; add straw and leaves and non poisonous yard clippings to their run, which will help cover the poop and allow the chickens themselves to mix everything up and make you some lovely compost for your garden. Having sun will help dry things out, so fewer flies, and your chickens will enjoy it on cold days. You can encourage them to scratch around in it if you toss some scratch or food out for them. They really are wonderful helpers.
Also useful are fly traps, the yellow sticky ones work best. I don’t use them every year, but some years flies are worse than others, it really depends on the rain.