Exposed Wood Framing Near Firebox
So the main issue is that the area isn't flat and to make it flat you do not want to chisel out the thinset over the board because the bottom row could loosen.
I would add some primer to the wood, thinset it to as flat as I can then install some metal flashing over that.   If the area is deeper than the picture you could just install some tiles that would go with your setup but I would want them generally out of view.
Also there is building code on combustables near fireplace.   You have to be further than 2&quot; - which this isn't.   I don't think any inspector near me would OK this so you might as well fix it, as you would probably have to when selling house.   Did your home inspector not catch this?