When part of the porcelain paint pops off of metal sinks, your only DIY recourse is to use the little bottles of touch-up.
First you have to find one that is the right colour, not always an easy task.
Then you have to get it to stick. The manufacturers all say to just clean the surface and then apply the touch-up. But in reality that never sticks for very long. They are just too afraid to admit that this is in fact a long job.
Start by using a Dremel or other small grinder and grind the black surface down to bare metal. Make the edge of the problem as sharp as possible, not feathered out.
Now apply the first coat of the touch-up. Let it dry thoroughly and then use very fine wet/dry paper to sand it down smooth, being careful not to scratch the surface around the patch.
Then put on another thin coat, let it dry and sand it down. It will usually take four to five coats to fill the hole since between each coat you remove a bit of what you put on. Mostly, the sanding is to take off the fat edge. (There do exist special "fillers" for this task and they do work, but this technique ensures that you have the colour right through to the metal.)
Despite all these efforts, these patches do not last long in the bottom of a sink. Follow this link to find out why.