I would have the fan checked to ensure it's been installed properly, often we come across fans that have been installed where the air is not actually venting outside but being recirculated back into the bathroom or into the void in the ceiling above the bathroom.

Hold a piece of paper against it, if it can support it without your hand helping then it
has suction . If it makes a faint clunking sound especially on start up the bearings are failing, or heavily linted up, (dust) on the fan. Good luck Steve Clayton

Most fan manufacturers recommend monthly cleaning of the fan blades.

Hi if the patch has only recently started apearing then it could be either that it is blocked at the fan entrance with lint and dust etc and needs a good clean, it could be that the outlet is blocked or has something lent on it in your loft starting a condensation build up or it could just need replacing completely. I would check the first two things and wipe down your ceiling with a mould cleaner, then possibly use a sprayable stain blocker paint, then wait for a few weeks to see if this solves the problem. If not call your plumber or electrician who will be happy to take a look. Thanks John