To get the mold off, you can try filling a spray bottle with pure bleach and spritzing an inconspicuous area first. If there is no color fading, go ahold and spray the moldy areas. This should kill the mold. As for preventing it, maybe a shellack?

keep out of shade.

they have sprays for that or mix a lite solution of bleach in a spray bottle

Bleach doesn't kill mold(Mould).

There are heavier solutions which will kill mold.

All Bleach does is remove the stain and send the mold into a dormant stage where it WILL re-appear and multiply even worse cause Bleach is nothing but water to the Mold.

Bleach will kill the mold for a time but will not keep it from coming back.

Go to Lowes or HomeDepot and look in the paint section. They will have a siding wash that is usually used to clean the outside of your house. It has an inhibitor in it that will keep the mold away for two or maybe three years. I use this all the time on our porch, rockers and swing.

Just follow the directions.

I looked this up on Google:
Any furniture or personal item that has actual mold growth, or primary contamination, should usually be disposed of, especially if the furniture item is soft or porous. In some cases of moderate mold growth on the surfaces of solid wood surfaces, the mold growth is limited to the top surface of the wood, and the item may be treated to remove the mold without damaging the item too much. The impacted wood should be washed outdoors with a strong detergent solution. Sanding will usually remove any remaining discolored areas. Application of a clear coat finish after treatment will further protect the wood by sealing the wood and limiting access to moisture and oxygen, which will inhibit mold growth as well as keep any residual spores from becoming airborne.

With other items that are actually supporting mold growth, it is usually not cost effective to try to treat or remove the mold. With antique items or items of great value, the cost of treatment may be justified. In these instances, a professional furniture restoration contractor should be consulted.

try tilex mildew. i use it for all my outdoor furniture.

Even if you can get rid of the mold, it's going to come back. Humidity and lack of bright light and lack of air circulation encourage mold.