I have been wanting to build a covered run, with a plywood floor coated with Black Jack 57 as is my new coop floor. I want to have an all weather run where the birds can go, rain or shine. I will still have an outside run, but my wants, are much like your condition. If it is not covered or you do not wish to consider doing so, I might just leave it bare, and wash it off as needed. I love concrete because of this and I have another coop with a cement floor covered with pine shavings.
It is good to just have a place to get them out of the coop, regardless of the floor material. You might want to look into placing a cover over it and turning it into an asset and not worry over things you can't change. With out good drainage anything you put on it will turn to muck. I suppose you could look into sand? I don't use sand, and some say it gets a stink to it, even with the poop removed.
A tough one perhaps,
RJ