Your spelling is good. You might try using a diluted solution of household bleach and an old toothbrush on the existing caulk to remove the mildew. If that fails, then remove the existing caulk down to the tub and wall surfaces. My favorite tool to do this is an inspection sticker scraper, available at auto parts stores. The folding handle affords good comfortable control of the cutting edge. Once you've gotten all of the old stuff out, make sure everything is dry-use a hair dryer if necessary. Recaulk using top shelf stuff-DAP and GE are my personal favorites. If you're not that adept at the art of caulking, there is a cheat trick. Using masking tape, apply a strip on either side of the joint to be filled, around 1/8" from where you'd like the finished line to be. Apply the caulk, using push application (pull doesn't fill the joint well) and then smooth the joint with your fingertip. Now remove the tape strips, one at a time taking care not to get spooge everywhere, and once they're gone, resmooth the joint, and it looks like you've been an extra on 'This Old Louse'.
Good luck