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The only way to kill mold and mildew and their spores is to use bleach. You can dilute it in a spray bottle with water, and it should kill the mold without harming the fabric. Try using about 1/2 cup of bleach per gallon of water. Test this on a non-visible area first. If the color does run, you can use a color safe bleach.
This is correct and a good reminder of how to tackle mildew stains on all kinds of surfaces.
For anyone who would like to avoid bleach, there are options. Unfortunately bleach will stain/whiten fabrics. And bleach is only effective against mold on non-porous surfaces - like tile (though some would argue that tile is porous too) and it's pretty toxic stuff. There are safer fungicides out there. e.g. Concrobium Mold Control.
One of the best things I have found to clean mold and mildew from my outdoor furniture and umbrella is bathroom tub and tile cleaner. spray on let ir sit a few minutes and rinse. It sure has saved me alot of money and made my things look new again !