I am with Shad on the ash - for bathing only and ideally not inside the coop.Bedding is for beds. Wood ash in beds is not such a great idea imo. Ideal bedding should be as dust free as is practicable. Chicken do not do well breathing in dust, much like us humans.
Bedding on coop floors is imo something else to be avoided if possible.
There are a number of reasons for this.
It's a waste of bedding.
It may encourage chickens to sleep on the floor when they should be sleeping on roost bars.
Straw, hay and some other bedding harbout mites.
Something all chicken keepers should get used to doing is inspecting poop in the mornings after they've let the chickens out. If the floor is covered in bedding it is very easy to miss thos spots of blood or runny poops that indicated a health problem in the flock.
I've read a few advocates of "the deep litter" method of chicken keeping. No chicken would be an advocate of it. It's a human concept that wasn't well concieved. Chickens left to their own devises are very clean creatures. Thye would prefer to roost off the ground, preferably on a high branch. This means when they poop it drops to the floor and all those bugs and stuff on the floor clean the poop up over time. Worms, lice and other unpleasant complaints are often picked up from chickens eat where they have pooped. Keepers should do their utmost to prevent a build up of poop anywhere chickens are likely to scratch and peck. A large pile of poop laden bedding in a coop is a prime site for scrathing and pecking and therefor a prime site for parasites.
Keep the bedding for beds. Put the woodash ouutside if you must for bathing in. Keep the coop floors clean of bedding and poop. Your chickens will be a lot healthier for it.
I have to say though that deep litter works well for me and effectively mimics the forest floor which I think is the chickens' ancestral home.
My Chicken Palace is on dirt and I just add leaves. Dig down in the leaves and just like in the forest floor you have lovely loamy soil full of worms.
Not everyone can do it that way though because it does depend on being on a dirt floor and yet still predator protected which I think is what I have managed to construct

I don't see them sitting or sleeping on the floor and frankly they dig around much more outside the coop than in it so I don't think they are digging in their poop any more than is determined by the fact that they are restricted and therefore not moving to completely new ground each day.
I also think I am lucky in that my density is very low so each chicken has a load of space (more than 200 sq ft each if you include the yard).
But we both agree, ash has no place in bedding in the coop!