About Broccoli

This sounds very much like white fly (the ones which affect brassicas). Personally I wouldn't use pesticides especially on a crop you are going to eat. In any case, I find that usually the plants outgrow the pest or natural predators get it under control. Unfortunately the predators you can buy for white fly in greenhouses won't survive for long outside.
I'd make life difficult for the pest by spraying the broccoli plants forcibly with a hose attachment; you'll be watering the plants at the same time which will help with broccoli.