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This may be blight, which is a disease that can cause your plant and the potatoes to rot. If left untreated, it will also spread to your other plants. Blight usually happens during the warm humid times of the summer.  Remove all diseased foliage. Treat the plant with a fungicide that is sprayed on the leaves. This is called potato scab and won't affect the taste of the potatoes. However, it will affect the storage potential of your potatoes, so you may not be able to store your crop for long. It is generally caused by poor soils and is best treated by amending the soil with organic matter and keeping potatoes regularly watered during the season. Slugs will eat the foliage and the tubers of the potato plant. You should be able to see the slugs on the plant, and their slime trails will be noticeable on the leaves. Bury a tuna can level with the soil and pour beer into the can, this will attract and drown the slugs. Slug pellets or Sluggo can also be used to kill slugs.
Look for dark blotches on leaves. Check for scabs that look like corky blemishes on the potato skins. Watch for slugs.