If you let the litter collect around the cactus, there will come a bug called cochineal. It looks like a white tuft of cotton stuck to the plant. That's the disguise. If you grab it and squeeze, your fingers will turn blood red. Anyway, these little things will kill the cactus for you but you have to have litter up around the cactus. If you have the litter but not the bug, go looking for it to transplant some to your land.
As for mesquite, you might be able to control young plants with goats. You can control the big ones with a tree eater. That's a machine that they drive up to the tree and it literally chews the tree into mulch in about 30 seconds. It's the world's biggest chipper/shredder all-in-one mounted on the front of a dozer. Here are some links