Powdery mildew on crepe myrtle
Yes, you should prune out and discard all of those infested sprouts at the base of the tree, to help prevent further spread of the disease. The lower young shoots of crape myrtles are more susceptible to powdery mildew. This disease develops in humid conditions. Pruning to increase air circulation is one way to help control it.
Take a look at the following pages for additional information about powdery mildew and how to manage it: