*squints* I thought that was a wall.
Well here's a lazy idea then... I assume the door swings outward? Get one of these white wire closet shelves (Home Depot, they're a couple dollars in the closet organization section) and do this, but in reverse, so the wire shelf "chicken gate" is sitting inside the run. Use bungee cords to hook it to the hardware cloth so it doesn't fall over.
It won't keep out fine particles but the chunkier litter will get held out of the door way, and the chickens will get held back at least temporarily. And you can just step over it to access the run. If you want to let the chickens out to free range, just undo the bungees and remove the "gate".
