Fungal diseases, while not common, can occur on leucothoe, especially if the plant is not in ideal conditions. The plant requires good air circulation so that leaves do not remain wet. High heat can stress the plants as can poor drainage. It is best to avoid overhead watering and instead water the soil beneath the plant.  

The best approach for you is to remove infected leaves when plants are dry, and remove any dead leaves or debris from around the base of the plant. Prune out any diseased stems or infected areas. This will reduce the overwintering inoculum capable of infecting new growth in spring. Be sure to clean off pruners between cuts so as not to spread and disease inadvertently.  Monitor the plant to see if it rebounds this season, or at least that the disease does not spread.  

Fungal problems are usually addressed in the spring by using a fungicide as a prevention rather than a cure. You will have to decide if you can manage the leaf spots through cultural methods  as recommended above, remove the plant for another species that is more suited to the site, or to try treatment with fungicide. Fungicides labeled for use on ornamentals with leaf spot include products with chlorothalonil, mancozeb, or thianophate. Be sure to follow label directions for dosage and safety precautions.