We just bought a new TT and the awning was horrible mess. So I just did this.
There are three types of awning material, canvas (which is not generally used), vinyl (common) and acrylic (expensive). My understanding is that vinyl will allow water and dirt (thus mold) to penetrate the fabric. Acrylic will only allow surface mold as it is sheet and not a weave. These should be treated differently. Gentle washing should do the trick with acrylic but will not do the job for vinyl. The process is that I use for vinyl is to wash with water and soap to remove surface dirt. I then use bleach in a pressure sprayer to soak the fabric. I leave it on for no more then 5 minutes and rinse with copious amounts of water. I leave it open to dry in the sun. This last step is important. Bleach does not last long when it exposed to sunlight. Leaving it in the sun makes sure that it is completely destroyed.
Now I am sure I will get a comment about using straight bleach but I do it for a reason. Mold does not form on a clean surface of the awning, it forms on the dirt that is there. If you want to remove the dirt that supports the mold you need to break apart the stuff (organic materials) so that you can wash it away. If you use very dilute bleach you will break apart the easy molecules (read colored) but not the bigger ones and mold will come back more quickly. Also the normal mode of deterioration of vinyl if called free radicalization. This is the same mode as bleach uses to tear apart molecules but the key is that vinyl used in outdoor applications have a ton of material that inhibits that from happening so it resists bleach very will.
If I just has a little mold then a dilute solution is probably best. I would still use it at 50% strength. I would not do the above every year either.
How do I know about bleach? I was/am a chemist that has used bleach in decolorizing applications in industry for about 5 years. It was an art to determine how much bleach and for how long to use it.