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Laying hardwarecloth underneath all the run is not the best plan in most circumstances IMO, unless you are trying to keep rats completely out (and even then, honestly, chances are pretty good rats will find a way in, there or elsewhere). Just do a good 2-3' wide apron, properly installed, along the OUTSIDE of the run fence, and you will be fine on digproofing. Brick laid in the run, btw, will NOT ratproof unless there is also hardwarecloth under it (and then, as I say, they will just come in elsewhere)
The problem with a hard surface (hardwarecloth, brick, stall mats, whatever else) in your run is that one of chickens' biggest strongest instincts is to scratch around and dig in the dirt with their feet. Mine *easily* go down 8" or more in their activities. You'd have to put a LOT of stuff on top of the hard bottom in order not to be interfering significantly with their life. Even if you don't mind interfering with their instincts somewhat, I wouldn't go less than 3-4" packed material (i.e. 3-4" *after* the fluffiness has been stomped out of the hay) just to protect their feet. (I speak from experience here, having some of my runs on concrete slabs just cuz the slab was preexisting).
The only reason I could see for using a hard surface in your run would be if it's unavoidably preexisting (like mine) or if you expect flooding problems. And frankly if you expect flooding problems, you'd be as well or better off (in the long run at least) FIXING the flooding problem -- by relocating the coop, by getting in a big dumpload of roadbase to make a raised pad, and/or by doing various water-diversion things. See my 'fixing a muddy run' page for mroe on the subject, link in .sig below.
Otherwise personally I think you'd be WAY best off with just plain dirt (possibly with sand or some other aggregate added on top of it).
You can TRY the "fill run with hay and remove it each week" thing if you want, but I strongly suspect you'll become dissatisfied and change your tactics pretty soon. That is going to be a lot of hay and work, and make a rather laaaarge long-lasting compost pile, and may STILL have stink and fly problems (especially if you do it over hardwarecloth rather than brick). Also, I don't personally feel this is a huge concern, but I will pass on the information that a number of BYCers have had chickens get impacted-crop problems from hay-bedded coops/runs... this seems to be infrequent and unpredictable, but probably likelier in chickens with a small boring run than in chickens who have a larger area and more things to do. So, I mean, by all means try it if you want, but it's worth looking at what your other options are going to be, too
Good luck, have fun,
Pat