Tile – Laying Tile Over Asbestos Floor Tile

I would probably use unmodified thin set to cover the gaps and give a good base for applying the tile.

Could you do something a little decorative and use a border tile around the repaired area, sort of like a picture frame? Then use a different tile to fill in the area. The border would make the change in depth harder to see.

Pry that tile up, it sounds loose on that corner - i.e. not only unsupported, but some of the thinset in contact with the tile did not adhere.

That looks like unglazed clay tile (aka mexican clay tile), which you must moisten on the bottom to get it to adhere well. If you don't, the tile sucks the water out of the thinset before the thinset can bond, and you get what you have with that "klonking" tile.

If it was just voids at the corners, you could just "pipe" some more thinset under the tile with a pastry bag, but after watching that video, I would pry up that tile. Hopefully you have some spares.