Repairing these stubborn depressions is important. BUT, it must be done right so as to not cause further damage to the greens. While some golfers fail to repair ball marks, there are many well-intentioned golfers who do “repair” the marks, only to do so improperly. You won’t go wrong if you follow these three steps:
Step #1: Take your repair tool and insert the prongs into the green at the edge of the ball mark. Note: Do NOT insert the prongs into the mark itself, but at the outside edge of the depression.
Step #2: Push the edge of the tool toward the center, using the repair tool in a gentle twisting motion.
Step #3: Once you’ve worked around the rim of the ball mark with your repair tool, pushing the grass toward the center, gently tap down the repaired mark with your putter smooth the surface.
For more tips on proper ball mark repair, see here: https://www.youtube.com/watch?v=GuYXMn4tA10