Are there burrows, or are the holes as far as the excavation goes? You could have burrowing voles (tunnels), but if there are no burrows, then I expect that a skunk (and/or raccoon) is digging grubs (such as cranefly larvae, etc.).

In that case, treating the lawn to reduce grubs would be one tactic. I'm a wildlife biologist, not a turf specialist, so submitting a new question here and asking "how to reduce grubs in lawn" would be one thing to consider. I know there are retail products that claim to reduce insect grub survival in lawns, but I am not familiar with the chemicals involved or other options - I believe some folks use nematodes, but again, those and their broader impacts are beyond my expertise. 
On the wildlife side, I would say besides reducing the attraction of this particular food source, you might want to 1)use or borrow a trail camera to make sure it's just raccoons and not raccoons + skunks and 2)decide whether having those animals use the yard is tolerable - once we interrupt that digging behavior. Exclusion for those 2 species varies according to the size of the area, whether (and what type/size) fence is present, etc. and 3)whether raccoons or skunks, look for any and all other sources of "free food" and eliminate those. Common sources of food subsidies to either of these includes spilled bird seed, downed fruit/garden produce, livestock feed (chicken to cows), fish pond, compost that isn't "hot", materials piles or garbage that might harbor tasty animals and invertebrates, etc.