Damage to the sub floor while scraping tile
If the damage is not significant (you mention the approximate area affected but not how deep the damage is) you could just tile right over it.

If the missing layer(s) of backer-board are significant enough so as to cause the tile to not lay level and flat, you will need to do something: 


best option would be to remove and replace the damaged backer-board
the Ditra product works well but also requires a flat level surface below, so again it would depend on the nature and extent of the damage.
I have had decent luck troweling a smooth skim coat over rough surfaces and letting it cure prior to tile application, and would probably choose this method because it is the simplest/easiest, but this is a skill. Well mixed mortar with the right consistency along with careful trowel work are needed.