Hardware cloth on bottom of run?

It will be hard on their feet and if nicks their feet can lead to bumblefoot.

Some people bury or apron out hardware cloth to deter digging predators (google hardware cloth apron coop).

 

I don't see any benefit to keep chicks from scratching. They will regardless what, except if you tie their legs. =D=D
Now tell us what type of predator are in your concerns. There are other options. But if you do want to put the hardware cloth down.. here are my views;; It being on the ground, it will eventually rust, and sharp wires will possibly protrude. Yes, I know you plan to use galvanized type. It will just take a little longer to happen. . If you still need to have it, I suggest that you bury it under , and have sand or whatever covering it 3 or 4 inches. This way if something is trying to dig its way into the run, they will find a barrier. If the hardware cloth is just on the surface, your chickens may , (I did not say will) hurt their feet. .. Sand is good to have for your chicks to dust themselves. Use the coarse type preferably . Also provides drainage when raining. ..
HOPE THIS HELPS YOU. WISHING YOU BEST.

 

I was going to put chicken wire down on floor but thought about them scratching & getting cuts so pulled it all up plus when I want to rake run that would of been painful too! Good luck & they will decimate any ground you put them on LOL. I made a 5x16 run yesterday w green grass today therr us no grass left haha granted the whole thing wasnt grass

 

You may want to consider hanging some 2 x 2" netting (heavy knotted- think snow load and hang it so it won't bring your fence down with the snow) over the area where they will be hanging out for much of the day. This saves my chickens as it gives them a place to rest and be a bit more off-guard during the day at times.

My netting is visible on the right, with some 2 x 4's just propped up under it (the thick wet snow that sticks to everything would probably rip where those post are and I take them down when I know we will get snow). Snow will bring down your fencing so I only attach it to fence posts. Even then sometimes it will rip a little bit but not as bad as that very flimsy bird netting that you are supposed to put over fruit trees. I got mine from Mypetchicken.com.

Oh forgot to mention that we have Bald Eagles and hawks all through the day every day hanging around. And owls. I still get losses sometimes.