Patching unwanted holes in glazed tile like this can be a tricky proposition. Here's a solution that may not always be perfect, but will make a big improvement. This is polyester resin, often used to repair car bodies. We'll mix up a small batch by adding some hardener and stirring it thoroughly. Using a putty knife, press the resin into the hole slightly overfilling it. Then just as the material begins to harden while it's somewhat rubbery, trim off the excess with a razor blade.
Next use 600 grit sandpaper to sand the filler smooth. Wetting the paper with water makes it work best. Now it's time to try and match the color of the tile. This porcelain touch up kit which is available at most home improvement centers comes with several different colors of paint.
The trick here is to mix these different colors together until we come up with one that matches the tile. Apply the paint with a small brush. You can test it on a nearby tile. When you get it the way you want it, then just paint over the patch and let that dry. Give it one final sanding with that very fine sandpaper and you'll have, well, maybe not a perfect match, but something that's a heck of a lot better than what you started with.