How do I take care of the spots on my peach leaves?

This is probably entomosporium leaf spot. It is a common leaf spot on many different kinds of plants. Retired UT Extension Agent Mike Dennison says several different fungicides will control the disease and he gives some options. Read and follow the label directions on any chemicals you use.
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Related Videos:
Spraying Peach Trees
Mixing and Applying Fungicide

Related Resources:
Disease and Insect Control on Home Fruit Plantings
Fruit Spray Schedules for the Homeowner