I don't clean anything out of the run.
Dry organic matter can be wood chips, dry leaves, pine shavings, etc. You don't want to use anything that will mat or pack down. A mixture of things is good.
I add a truck load of wood chips annually. The chickens organize it from there.
Don't bother with sod or trying to grow grass back in there. The chickens will just destroy it.
I would add a thick layer of dry organic matter for run litter. Whatever is available to you. I like wood chips because I can get them for free.
You can set up misters if your chickens seem...