Experts and BHG readers answer.
Save the Vegetable Garden
Once you get a heavy population, it's might tough to keep them under control. I would not keep spraying insecticides, unless you switch to something a little more benign, such as insecticidal soap. The key, in my experience, is as much about how well you cover the entire plant -- all surfaces, upper and lower -- as it is about the product you use. So try insecticidal soap, do a REALLY good job of covering the plant and then follow up in week or so with a another treatment.
Also consider using sticky traps. They're little glue-covered cards you put out in the garden and they trap the little critters. They'll never eliminate a population, but if you start early in the season, they can help suppress them quite a bit.
The leaf yellowing and white patch you describe is undoubtedly mildew, which is pretty standard on cucurbits by mid to late summer. It can be treated, I suppose, but it's pretty much the way it goes each year. Can be much worse in some years depending on heat and humidty. Typically, you wll get a crop, but then the vines may just fade away. Not worth tyring to preserve them, IMO.