Adding Layer of Caulk to Tile?
Remove the old caulk completely, with a razor blade or similar. The wrong type caulk was installed. Silicone caulk specifically made for bathrooms has a mildewcide in it which slows or stops mildew growth. That is what should be used.
There are a number of ways to smooth that caulk bead so there is no place for water to pool. I just use a finger, but some used a wetted finger, spoon, another type of curved tool, etc. Kepp a rag or paper towel to keep wiping the tool so the caulk does not pile up behind it. But don't put in too large a bead.