Step 1: Cut Out Damage
Cut out a rectangle of sod around the damaged area. Cut a generous area, or the sod will fall apart as you lift it out. It might help to stand on a board and cut against it to get a straight edge.
Step 2: Undercut Sod
Undercut the rectangle with a spade. Try to cut the sod to as even a thickness as possible, with at least 1 in (2.5 cm) of soil.
Step 3: Break Up Soil
Lightly fork over the newly exposed soil to break up the surface and to encourage rerooting. Add soil if necessary and firm well.
Step 4: Turn Broken Edges Inward
Lay the piece of sod again, turning it around so that the broken edge faces inward, and the cut edge aligns with the edge of the lawn. Check the fit and level, and make any adjustments. Butt up the edges, pressing down the replaced section firmly.
Step 5: Top Dressing
Fill the hole with top-dressing and work it into the seams to help them knit together. You can reseed the hole if it is large, using an appropriate grass seed. Water, and keep an eye on the repair until the grass has started to regrow, weeding and watering if required.
Step 6: Keep it Neat
To maintain a neat edge, recut it whenever it starts looking untidy; this is easiest when the soil is moist but not wet. Lay a plank on the lawn to stand on and cut against, or use a taut length of string as a guide. Sever the grass roots with a spade or edger where they are spreading into the bed or border. When you have finished working in one direction, work back in the other, pressing against the cut surface to firm it.
Summer can leave lawns hard and patchy. Try these tips for Improving a summer-damaged lawn.