If you do decide to stay, then you can clean the mould but it will take a few weeks.  This is what I've done with our tiled shower after lots of frustration:  It is not as simple as using Exit Mould or whatever, all those products are just bleach which just removes the colour.  You need to use 80% vinegar, 20% water in a spray bottle and frequently spray the mould especially after each shower.  Better still, but I don't have time for that, is to put damp paper towels over the mould and keep it wet with vinegar.  It tricks the mould into overeating and then the cells literally burst, killing it.  Then you have to scrub it off (after a week or so).  It will stay black, but will come off easily.  Then repeat the process, spraying for a week or so.  Once you are satisfied it is gone, and if there are some left you can't scrub out, then use Exit Mould.  From there on, use 20% vinegar as a maintenance spray.  Consider painting a sealer over the grout to prevent it from taking root again.

Long term solution for painted surfaces is to use a mould resistant paint.  The one we've used in our old home on the bathroom ceiling had this oil oozing out of it for some time, but never had any mould ever again.  Also no bugs as the oil attracted and killed them ;-).  We did not care much for the old home and actually only used sugar soap and then Exit Mould and then painted, and the anti-mould paint killed off the mould.  Bought it from Bunnings.