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It sounds like you need two things:
1) much smaller-mesh wire, at least at "chicken level". This means, the bottom 2-3' of the run fence, plus anywhere within several feet of a roost or other object they can get up on. Basically, anywhere a predator arm reaching thru the mesh could POSSIBLY contact a chicken, EVER. 1/2" hardwarecloth is probably your best bet for those purposes, especially since you have a longstanding problem.
and also
2) digproofing. Some people would bury the base of the fencing in the ground, but you really need to go at least 18" for good security and that is a LOT of work (and the buried mesh will rust out, yes even if it's galvanized, and you won't necessarily know it's got too weak til a predator finds it out FOR you). COnsequently my preference is for an apron of wire mesh laid on the ground on the outside of the run fence, securely attached to the base of the run fence along that edge and pinned WELL down to the ground on the other edge. You can shovel earth or mulch or gravel over it, or cover with rocks or pavers or concrete rubble, or just pin it down very well and let the grass grow up through. But for good protection it should be 2-4' wide -- given your history I would suggest closer to 4 than to 2 feet! For most situations, realistically you get about as much benefit from a good heavy-gauge galvanized 2x4" mesh wire as you do from anything smaller. I don't personally see any point in using anything smaller than 1x2" unless you are trying to keep out weasels and rats, but that is SUCH a difficult project that they will probably find a weak place elsewhere, so if weasels and rats are around it is really just better to lock the chickens in a very secure chickenhouse at night, IMHO.
(Note that both of the above things apply to the the building as well as the run!!)
You cannot bury an electric fence wire -- well I mean, you can, sufficiently-insulated, but then it is not ACTING as an electric wire, it will not zap digging creatures or anything like that.
In your situation, you might choose to add one or two hotwires to your run fencing, but honestly it sounds like you have significant fixes to make ON that run fencing first -- it is really not a good idea to try to compensate for a grossly-insecure run fence by adding electric, because all electric fails sometimes, and predators notice real quick, especially when you've unfortunately been feeding them for the past three years and they KNOW it's worth gettin' in there. Honestly I'd suggest you not worry about electric and just get the PHYSICAL ffence to where it needs to be, to be reasonably predatorproof.
Good luck,
Pat