I would! Just mix in the coop cleanings with other plant sources, like leaves, straw, kitchen waste (no meat, eggs or dairy) , grass trimmings (if no herbicides/insecticides used) and etcetera. The compost works faster with chicken coop waste included and likely contains more Nitrogen in the finished product. Diversity of materials is good! My gardens love it! I do not add coop waste to my red worm composting bin, only fruit and vegetable/plant waste, then with no onion, citrus, manure or meat/dairy. The worms hate citrus and onions and the meat/dairy creates flies, rats and potential human disease issues into a composting project. I think the wood ashes mixed with chicken waste will, when mixed with a diverse plant material base, make a fine composted fertilizer! While I never add wood ashes to my compost pile, I think when mixed with chicken coop waste I would give it a go! Just my opinion, no science. Best wishes for you!