I doubt this is a cheep fix,, where you gonna find a tile to match..Your going to have to grind out mortar, mastic a  new one in that I doubt you will find a match for it and then re grout..it's in a worse than awful spot on an inside and in a corner.. you'll probalby have to take several tiles out..tile work is tricky once you start to force one tile the others chip..  

As far as other materials and tools dry wall putty (touch up paint) caulk Roofing MASTIC <--- this is a must for roof maintaining Also bring garbage bags and a long hose if you dont have one.

On some of the units, the bottom layer of flooring had vinyl tile with traces of asbestos and in many of them the mastic had asbestos. 

Our company uses Mastic Ply Gem technology to create a vision for what the different types of siding would look like.

Some kind of quality mastic would probably do just as good of a job and cost way less.

White cloth tape at ductworks, mastic, 9x9 vct tile are items to look for that contain asbestos.

But I use mortar, not mastic to install tile.

Property is in Mastic Beach, its in a C neighborhood.

Recently we pulled all the paneling off the walls in the basement of a house we were redoing and as mentioned earlier there was a nasty mastic holding the paneling on that was giving us grief to remove, so instead of spending days scraping we just put a new layer of 1/4 inch rock in its place and finished it out.

Only problem is that the tile was laid over black mastic which contains asbestos.