Thank you for bringing in clippings of your persimmon tree to the Yulee satellite office. The white stuff is an insect called scale. Both soft and armored scale can be pests of persimmon and other fruit trees. I suspect your scale is one of the armored scale species as we are seeing no honeydew or sooty mold. Those armored scale insects typically interested in persimmon are Hemiberlesia rapax, Greedy scale and White peach scale, Pseudaulacaspis pentagona. The photo I have added is White peach scale which looks very similar to your problem.
All scale insects are controlled by the same method: monitor the tree and look closely at the scale. When the eggs and crawlers (immatures) are present use insecticidal soap, horticulture oil or other insecticide in two applications about 10-14 days apart. This is not a preventative method; the insects must be present for the chemical to work. Dormant oil will also control scale but generally is only used when temperatures are cool. You need to apply something now. As always, it is critical to follow the directions on the pesticide label.
A few other important hints about fruit trees to consider. Remove lawn grass as far away from the tree as possible. What we do to lawns we should never do to any tree whether it is a fruit tree or an ornamental tree. Be sure mulch is not piled up against the tree trunk. Often fruit tree bark is thin and allowing mulch to accumulate around the trunk can easily contribute to disease entering the tree. Do not over water. Once a week is sufficient in the early establishment of the tree but afterward, if we receive sufficient rain, the tree many not need to be irrigated. In addition, over use of high nitrogen fertilizers can cause the tree to be just too attractive to insects.
Lastly, please consider purchasing persimmon varieties suitable for our climate. Astringent cultivars will not have a pleasant flavor unless they completely ripen on the tree and then they become extremely soft and mushy. Non-astringent cultivars can be eaten when fully ripe or while they still have an apple-like crunch. Both types of persimmons will grow and fruit well in North Florida. Oriental persimmons (astringent & non-astringent) are known scientifically as Diospyros kaki. The native species D. virginiana has smaller astringent fruits.
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