Lawn top dressing
The purpose of topdressing a lawn is to maintain the levels of the lawn and to help the grass plants to produce more shoots maintaining a denser thicker lawn. Top dressing is carried out by working in some topdressing material. Ideally this will have been created from some local turf that has been rotted down to create a loam with some sand added.
In your case once the turf stack has rotted down we will pass the soil through a screen to remove and stones add some sharp sand and work the resulting material back into the lawn as a topdressing plus some grass seed if required.
The key is to apply as much topdressing as possible but at the same time not so much that it smoothers the existing grass allowing it to regrow through the top dressing.