Mold that forms on wood is generally just on the surface. If the mold does go deeper into the wood, it likely needs to be completely replaced. If you do have surface mold you can try to attempt to kill the mold and clean the wood.
If you try to wipe the surface while it is dry, you will send the spores of mold airborne and risk them populating another surface. Once you have the appropriate safety gear on, gently spray the affected area with a light mist of water.
In your bucket, mix a solution of 1 part detergent, 10 parts bleach and 20 parts warm water. Use a scrub brush with bristles and wash down the pre-dampened wood.
Make sure there is very good circulation for the wood to dry quickly and completely.
Please see links below on identifying mold and information from the CDC and EPA.