Persimmons will drop fruit when stressed and it seems that something is stressing your tree. Usually this is due to a root problem. Dry soil, poorly drained soil, a planting hole in clay soil that has slick sides preventing roots from venturing out into the surrounding soil. The solution to these is to keep the soil moist but not soggy. Mulching helps. There is little to do for a 5 year old plant if the planting hole was the the problem. In time it may fix itself.
Another common cause of a lack of vigor and fruit is shady conditions. Persimmons need good sun exposure to perform their best.
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