Lacebug, not lacewing. Lacewings are very beneficial predators of common pest species. Lacebugs are the pests.

It's doubtful that mosquito abatement applications do anything to control insects that live UNDER the leaves. They do, however, kill beneficial predatory and parasitic species.

Lace bugs are soft bodied insects, so any contact insecticde will be effective if sprayed directly on them. Very low toxicity products such as insecticidal soap, bioNeem, etc. are effective but have no residual action; this means timing is critical to hit the first generation. There are other products which do have residual action, including systemic soil drenches containing imidalcoprid. The long term effects of the last one, particularly on bees, is a growing concern.

As rhizo mentioned, lace bugs live on the undersides of the leaves. Any spray needs to be sprayed from the center of the plant outward so it hits the underside of the leaves.

The way to avoid lace bugs is to only plant azaleas that don't get lace bug in the sun and plant azaleas that do get lace bug in partial shade. The natural predators of lace bugs are the best defense and they don't like full sun. So if you plant azaleas in the right spot, they are trouble free. If you don't you are inviting problems. That is true with any plant. That is why it is so important to read the labels to get the cultural requirements of plants when you buy them.

Here are some azalea varieties resistant to lace bug:

From Oregon Sate University Extension

Indica alba (white), Flame Creeper (white), Delaware Valley White (white), Rosebud (pink), Cooperman (red), HahnâÂÂs Red (red), Boldface (lavender), Mrs. G.G. Gerbing (white), Dream (white), Salmon Beauty (pink), and Hinocrimson (red).

North Carolina State University Extension

Fourth of July, Koromo Skikibu, Autumn Amethyst, Autumn Twist, Autumn Royalty, Autumn Sangria, Autumn Cheer, Autumn Rogue, and Gumpo White.

Clemson Univeristy Extension

Dawn, Pink Star, Ereka, Cavalier, Pink Fancy, Dream, Seigei, Macrantha, Salmon Pink, Elsie Lee, Red Wing, Sunglow and Marilee.