Damaged areas, especially on the edge of a lawn can be difficult to repair. You can of couse cut it away, if there is plenty of good lawn area. Use a string line or some other guide to get a straight edge, it is almost impossible to maintain a straight line over a long length.
If the damage is severe and on the edge, you might consider cutting a square section into the lawn, as shown below, cut away the top lawn surface, build up a solid base of soil right up to the edge , if necessary support the edge with some metal edging strip. If you are using a heavy petrol lawn mower, the edge of the lawn will have to support its weight when mowing, if it is uneven, or prone to break up, you could end up with a bald patch again!
Turn the removed area round, so the damage faces INTO the lawn, away from the edge, fill in the gaps and damaged areas with well rotted compost and reseed to complete the repair, watering well.