Because the soil is soft at this time of year and the growing season has begun, this is a good time to press mole-raised turf into place. If the damage is very extensive, using a lawn roller may be helpful but keep this activity to a minimum to avoid compacting the soil.
Unfortunately, when conditions are favorable, moles will probably return to do what they did last year. Trapping is the best way to reduce their numbers but in the long run it's probably necessary to tolerate some damage. When moles are trapped and removed, others usually move in quickly to occupy the vacant tunnels.
The following bulletin discusses ways to reduce mole damage. Some methods are more effective than others and it pays to concentrate on those that work best.