Most people on this forum, me included, use "hardware cloth" to refer to welded wire mesh that is 1/2x1/2" or 1/4x1/4" -- implication being that you are GOING to use a galvanized version, although some versions are certainly *better* galvanized than others.
Whereas "welded wire" is used to refer to larger mesh sizes, from 1x1" on up; these are almost-invariably in heavier-gauge wire as well, and I will go out on a limb and say I'm not sure you can even FIND them in non-galvanized, although again there are certainly better and worse qualities of galvanization (and welding, and gauge)
To some extent it is a regional thing though and not everybody on the planet uses this same terminology.
By an apron, I mean take some good quality fence wire (what you are using for your fence is ok, or something vaguely comparable) that is 2-3' wide, and lay it flat on the ground outside the run fence all around the run. Attach it *securely* to the base of the run fence. Bend the free edge down a little and either peg it down real well and let turf grow up thru it, or weight/cover the wire with "something" (mulch, gravel, rocks, concrete rubble, pavers, whatever). You can also do this by using extra-tall run fencing material and bending the bottom 2-3' of it outwards at a right angle to create the same effect, but in most cases that leaves you with a shorter-than-desired run fence *and* is a bit of a nuisance.
Good luck, have fun,
Pat