What to do with a hole beneath my lawn

All you can do is fill in the hole. The rodent will probably have moved on to a new burrow. If you want to be really nasty you can put some poison bait in the hole. But that little rodent may well be eating some of your other pests!
The only rodents where I draw the line are coypu, as last autumn they came out of the pond in our garden and ate twelve beautiful red cabbages down to the stumps! Be thankful you only have a few field mice or the odd mole!

i dont think the UK get many coypu! A good thing if they eat everything!