Wood chips are the best! That's what I have (plus other plant material mixed in, whatever is available seasonally - lawn clippings, yard waste, dry leaves etc.) I don't clean the run. The chickens mix it around when scratching, and it composts. The wood chips take years to compost, but they help the poop and the other plant matter get ground up and composted when being scratched through and mixed.
One thing stood out to me from your post - chicken wire. It would be good to look into replacing that with something else, as chicken wire can easily be chewed through by predators. A lot of the common chicken predators can climb, and will have no issue reaching your chicken wire and getting through it. Have you had any winters with your chickens yet? Chicken wire also collects snow - the holes aren't big enough for the snow to fall though, especially heavy, wet snow. So depending on how the frame of your run was built, and how sturdy it is, the collected snow may weigh it down and collapse it. If you think your structure can withstand the maximum snow load you may get in your winters, then the best thing to cover the run with would be a welded wire or hardware cloth mesh that will stop predators from getting in. If you're not sure that it will withstand the snow load, then get something with larger holes - like 2"x4", but still welded wire. That way smaller things like weasels can still get in through the holes, but at least you won't be getting any big things chewing through, like raccoons. And make sure the chickens are locked in a fully predator-proof coop at night, so the smaller things that can slip through the run top won't be able to get into the coop itself.