For my
experience with AP systems, treating nuisance insects is similar to the way you would treat them in most annual vegetable gardens.
As Casie said, you can put a fish in the grow bed that loves to eat mosquitoes. I think that may be your best option for the mosquito problem. Just make sure that they particularly DO NOT like to eat plant
roots. We had frogs naturally establish and eat all of our mosquito larvae and we were so excited...until we found out that they ate ALL of our plant roots, too!!
For white flies, try to get more air flow/circulation in the AP system area if possible. More air flow around your plants would, I think, be better for controlling some of those things like white flies. You can also apply organic pesticides. I know it is not ideal in a
permaculture sense, but if you are growing annual vegetables in a system that is quite manufactured (like in most AP systems), that is one of the best SHORT TERM solutions. Be very careful and sparing with how you use the organic pesticide. Obviously, you are dealing with aquatic life, including fish, so it presents more problems. If you do spray, try your best to only hit the plants directly, with as little getting blown/dripped into the
water as possible. Of course, if you make a homemade pepper spray or something of the sort, it may be a bit safer, but anything with an oil or other surfactant in it will be bad for the fish (and other aquatic life) if too much of it gets into the system.
Looking on the longer term side of things, I would definitely plant as many plants as possible that help deter certain insects...different herbs, etc. The benefit there is that you can harvest many of them. Also, if the insect issues are not too bad, you may just be able to be patient and let things work themselves out. That depends on if you are
gardening indoors or outdoors as well as other factors, but letting natural predators come in just like you could with a regular annual veg. garden could work.
For the "other" flying insects, I would try to identify them before doing anything to prevent them being around. Make sure that they are detrimental and not "beneficial" to your cultivated plants.
Again, in my experience you can treat it like a regular annual vegetable garden with extra care taken for the aquatic life in the system. Start with the least invasive or drastic option and go from there.