I add pretty much any and all organic material and vegetation into my chicken run. It's basically like a giant compost pile and is wonderful topsoil now that's it's been established for a few years, I use it in the garden and in my potting mixes, every shovel is loaded with worms and soil life.
I'd start with a truckload of arborist mulch so you have a nice thick base layer, then add in all the other stuff: leaves and rakings, mulched up plants and trimmings and grass clippings etc (I just run them over with lawn mower to chomp them up smaller before dumping in the run, just make a pile the chickens will spread it out), food scraps, old bedding, shavings, hay/straw bales, you name it.
Since you mention drainage - over time you will be raising the terrain inside the run with all this additional material, above the natural terrain - so standing water and drainage shouldn't be an issue...at least inside the run