To deter digging predators, either bury wire fencing 18" or more deep at the base of the run fence, attaching it securely to the base of the run fence so no gap can be made there; or run an 'apron' of wire fencing horizontally on the ground (or just under the surface of the ground), from the base of the fence out to 2-4' away (in your case I would go all the way out to 4'!), again affixing very securely to the base of the fence. This needs to be heavy-gauge galvanized wire, no larger than 2x4" mesh. If you do an apron on the surface of the ground, pin it securely to the ground using tent pegs or heavy rocks/pavers/concrete-rubble/whatever (and preferably turn the far edge downwards).
To keep things from coming in the top, a well-built roof is the most effective but also least-easy/least-cheap option. If you do build a roof, build it RIGHT, not in a half-*ssed way. If you are not going to do a roof, heavy gauge galvanized wire mesh on a STRONG wood frame with lots of 'rafter' type crossbraces will work well too. If baby possums and baby raccoons are an issue, or snakes, then you want 1x1" mesh; otherwise 2x4" mesh can be ok.
Make sure that everything is strongly and tightly built, with no 'oh, they won't notice this part' gaps or wibbly bits or insecure attachments.
It is a real good idea to ensure that the bottom 2-3' of your run fence (also the top, if your run is less than 4' high) has 1/2" mesh on it, ideally hardwarecloth but 1/2" chickenwire (hard to find - the usual stuff is 1") or even 1/2" plastic garden netting are somewhat better than nothing. This will prevent reach-through problems, where a chicken sticks its head out thru the fence and gets injured or grabbed, or a raccoon reaches in to grab a handful of a nearby chicken. You might think the latter would not happen, but it is actually a fairly frequent way of people losing chickens.
If you are having predator problems, btw, DON'T USE CHICKENWIRE at all, for anything. It is jsut not strong enough these days. Some flimsy square-mesh welded wire and chainlink fencing are also on the market. It is really worth saving your pennies, or forgoing cable or cell phone service for a while, or stopping eating out or buying new clotehs for a few months if necessary, to build a GOOD STOUT run and predatorproof coop. And of course lock them into the coop every evening at dusk, without fail.
Good luck, have fun,
Pat