Answer: While a fungus can kill your plants, leaf miners rarely do enough damage to harm most plants. Leaf miners are the larvae of either moths or flies. The adult lays an egg just under the leaf surface. When the egg hatches the larvae feeds and makes tunnels in the tissue between the upper and lower surfaces of the leaves. They are protected from pesticides because they are within the leaf. When they're ready they make an exit hole, spin a web and lower themselves down to the surface where they pupate and turn into adults. To try and control the next generation, rake the soil beneath your plants to expose them to the weather and to predators.
There are some fungicides registered for use on vegetables so check the garden center for those you might be able to use. It has been my experience that once a fungal disease takes hold of a plant, it's difficult to control. Let's hope for the best with your plants.
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