When a tile repair is needed often it is the tile that is loose or moving. The movement is causing the grout or tile to crack or move.
Alter the tile is replaced, adhered and dry the grout can be applied. You have a choice to use the existing grout or use grout caulking.
They make both sanded or non-sanded grout caulking in most grout colors and these can be a used in lue of grout and allow the tile to move a bit without cracking or pushing pressure on the tile. It is often a good solution if movement is still an issue.
Use non-sanded for 1/8” or smaller areas.
Use sanded grout for 3/16” and larger grout joints.
I hope this might help you do a correction that last.