IMO, bare ground, be it in the garden, or in the chicken run is unhealthy soil. By adding a covering, it protects from erosion, and protects the beneficial organisms so they can work better. A deep litter in the run will encourage lots of tasty worms and insects to move in, to feed those chickens and give them plenty of stuff to do! My garden and run devour every bit of organic material my yard and flock produce. So, I go looking for extra stuff to bring in. Every trip to the dump, after dumping my trash, I load the truck up with cardboard. I have wood chips delivered by the 14 cu. yd. chipper truck. I have 2.5 loads left to spread. This weekend, I plan to go to the dump, time allowing, and park my truck at the drop off site for compostables. 1/2 hour of parking there, should yield a full load of already bagged leaves. I'll hope to get at least 3 loads to bank around the coop and green house, put into the run, onto the garden. Leaves are black gold.
In the run: when you clean the coop, toss those shavings into the run. Mow the lawn: toss the grass into the run (assuming you don't use insecticides or herbicides). Leaves: in the run. Kitchen, garden scraps: into the run. Aim for a 6" layer of organics in the run. Good luck with that! Those beneficials and the chickens will devour the stuff. What the chickens don't eat will just melt into the soil!