Repeated infestations. Yuck.

There's an unidentified shrub in our backyard that's subject to repeated infestations of tiny white parasites. I've sprayed it with a general insecticide -- this knocked it out last year, but seems only to have slowed things down this year. (We've had strange weather -- mums and magnolias flowering at the same time; middle Tennessee, Zone 6.)
I'm not sure what to do. Not knowing what the plant or bugs are is a big part of the problem. The other is that I don't know if this shrub is appropriate where it's planted -- does it need a lot of sun? Dry soil? Moist?
Should I take it out and plant something more pest resistant? Is there hope for saving it? If I don't take it out, how can I prevent a similar infestation next year?
Pictures are here:
Any help would be _greatly_ appreciated. I'm still learning how to manage what the previous owners of the house planted....
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