Hi Jack –
There are a few ways to control the fig whitefly. Some options are a soil drench (systemic) insecticide, a foliar spray insecticide, or using natural predators or beneficial insects. From reading recommendations on how to get rid of the fig whitefly, I will first suggest looking into biological control.
Natural predators are the best bet. Insecticides can become less effective over time, and since the fig whitefly reproduces quickly, I believe it wouldn’t take long to start seeing a loss in effectiveness. Some natural predators are lady beetles, a wasp parasitoid, and green lacewings. Systemic insecticides are often recommended, but most are not safe and can be harmful to bees. As for foliar sprays, I recommend neem oil, Botanigard, Horticulture oil, or Insecticidal soaps.
Hope it helps!