I've got a tiled kitchen countertop and have noticed some mold under the caulking around the sink, especially where the tile dips.
Should I clean out the old caulk and re-caulk the area, or pop the sink out then clean and start over?
Never caulk an overmount sink into place. If you've done it right, you will have to destroy the countertop to get it out if you have to remove it down the road. Silicone is incredibly strong when used correctly.
First, remove the sink. Then spray bleach and water (50% v/v) all around the mounting flange and the hole in the counter. Let it sit for a few hours and repeat 2 more times for a total of 3 times.
Second, get yourself some stainless plumbers putty. Roll it into thick ropes (1/2" dia) and place it all around the bottom side of the mounting flange. It should form a continuous ring underneath the lip of the sink. Just join the ends together by squishing them into each other.
Third, reinstall the sink and tighten the mounting screws a little at a time all around the perimeter of the sink. Trim the excess putty away from the sink flange on the top of the counter once it's done.
Pictures always help with these questions by the way. No big deal though. Always use a caulking labeled 100% silicone and mold resistant for this application and being that you have mold already there its not a bad idea if you have the means to remove the sink and clean the area below the mounting where you can't normally access for cleaning. Then reinstall it after it is completely dry of course if there had been any moisture trapped in there. If it is an over mount sink you put caulking underneath of the rim which is what I would recommend. Make it thick enough to where you know it will seal any water from slipping between there and of course keep the bead continuous with no breaks and it. After putting the sink back down make sure to use the clips underneath all the way around evenly has to apply pressure to the mounting and squeeze the caulking evenly.
I would recommend to scrape all the caulk then spray bleach solution, let it dry for 24 hours the caulk with a good and mildew resistance caulk.