Mold grows when you leave a damp item in a dark, damp area. Vinegar kills mold and won't discolor a cork board like bleach. Vinegar also kills odor from mold. The vinegar smell disappears after drying, so don't worry about a lingering vinegar smell. Prevent future mold from growing by storing cork boards in a dry place. Occasionally spraying a cork board with vinegar also helps prevent mold.
Remove all tacks and other items from the cork board.
Fill the spray bottle with vinegar.
Spray the mold with the vinegar.
Use the soft cloth the gently wipe away the mold. Cork crumbles so don't press hard or scratch at the board.
Spray the board again and let it sit out to dry. The vinegar soaks into the cork and kills deep mold.