Honeydew is a sticky, sugary liquid that... is insect waste. Insects called aphids (and some scale species) suck on plant sap and secrete a liquid we all know as honeydew.
For the most part, an infestation of these messy insects won't seriously harm your tree. However, they can cause distortions of leaves and of fruits and flowers. Which can be aesthetically unpleasing or a big problem if you grow the tree specifically for it's fruit. There are preventative measures that can be taken against the pests if this is the case.
Another byproduct of these sucking insects is sooty mold. Sooty mold is that black stuff that shows up on the bark and leaves of some trees such as crepe myrtles. The mold forms on the sticky waste of the scale insects. Therefore, get rid of the bugs and you get rid of the mold.
Oils, insecticidal soaps, and some synthetic chemicals such as malathion are effective for controlling aphids and scale when used and timed properly. However, we don't typically recommend foliar insecticides because it can disrupt the natural balance of the ecosystem by killing beneficial insects. This can create other problems on your tree and in your yard.
To treat it, we recommend two methods. 1) Apply a systemic insecticide at the proper time. Imidacloprid is the recommended ingredient, and many manufacturers sell it, including Bayer and Bonide. Find a formulation that you can apply as a soil drench, which is the safest method of application. 2) Use a high pressured water hose or a wet rag to knock the insects off your tree.
We are based in Highland Village, Texas and primarily serve Denton County and the North DFW areas: Argyle, Bartonville, Carrollton, Copper Canyon, Corinth, Denton, Double Oak, Farmers Branch, Flower Mound, Frisco, Justin, Krum, Lake Dallas, Lewisville, Ponder, Roanoke, Sanger, Shady Shores, and The Colony