Ask Mr. Smarty Plants

Generally, the Malvaviscus arboreus var. drummondii (wax mallow) is a pretty tough, well-adapted plant for our area, and not much bothered by pests and diseases. There are three insect pests that can disturb it, creating various symptoms of the presence of the pest, one of which might be the "cotton" you are seeing on the leaves. Go to the websites indicated to see the symptoms, matching them up with your observations, and find a treatment that seems reasonable to you in the situation. Given the area where you are seeing this cotton, we would suggest that you trim back the area of the Turk's Cap that seems most affected. Since you have already observed that it is worse in the most-shaded area, and since you can prune Turk's Cap nearly all year long and not damage it, we would certainly suggest that you trim that out of the shady area, and dispose of it so whatever bug is causing the problem will not get an opportunity to spread somewhere else. 

Aphids: these will definitely cause leaf curl, and the honeydew they excrete begins as a whitish area, although it can develop a black fungus that will change the color and is really ugly. University of California Integrated Pest Management Aphids. 

Spider mites: this is probably the best suspect in your case, as the spider mite is a member of the Arachnid (spider) family and has web-spinning abilities. Again, from the UC IPM program Spider Mites. 

Mealy bugs: these individual bugs can also have a cottony appearance, and you can find some easy control methods in this eHow site How to Control Mealybugs.

From our Native Plant Image Gallery:

Malvaviscus arboreus var. drummondii

Malvaviscus arboreus var. drummondii

Malvaviscus arboreus var. drummondii

Malvaviscus arboreus var. drummondii