Answer: What you describe could be either mealybugs or cottony scale insects. Scales for the most part are sedentary creatures which are difficult to control ? especially with sprays ? as most have a natural talent for self-preservation. First of all most of these pests, at least as adults, permanently attach themselves to the plant or tree which plays host to them. Secondly, most are equipped with a protective covering, usually composed of waxy secretions forming either a shell of sorts or dense mat of waxy filaments or fuzz. These characteristics contribute to them being very troublesome and difficult to control.
Mealybugs also cover themselves with a protective layer of waxy filaments and they, too, are difficult to control with insecticides. I've had good results through the laborious process of dabbing each cottony mass with a cotton swab dipped in rubbing alcohol. This effectively disintegrates the protective outercoating and kills the pest. You may have to treat your jasmine several times before you get all the pests.
Or, you can prune out most of the infested plant parts and then dab the rest. Either way, you'll need to control the population before your jasmine can regain its health.
Best wishes with your jasmine!
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