The insects you describe are almost certainly not termites, which do not feed on lawns. One possibility is sod webworms; they make tiny burrows in lawns from which they emerge to feed, mostly at night. Regardless of what your insects are, a general-purpose insecticide, Sevin, for example,will probably eliminate them. However, it’s worth a trip to the cooperative extension service to find out for sure what the problem is. Take a sample of the damaged turf and an actual specimen of the bug, if you can find one. Experts will be able to identify it and make a recommendation.