Answer: Although it appears you are asking about vegetables, there are in truth none I can think of that could thrive in such conditions. Most of them require at least six hours a day (including noon) of sun and would have trouble coping with tree roots and the inevitably dry soil under trees.
More important than the pH of your water would be the pH of the soil. In a shady spot with clay soil you might try hosta (available in many foliage colors and textures and patterns as well as some with fragrant blooms) and periwinkle (vinca minor) both of which are very tolerant of a wide variety of conditions. When you prepare the planting area be sure to incorporate plenty of organic matter (compost, old rotted leaves, aged stable manure and bedding and so on) into the soil and then keep the plants well watered and mulched with an organic mulch such as shredded bark or rotted leaves until they become established. Your county extension (270-2781) may be able to recommend additional plants known to do particularly well in your local area under similar conditions.
I'm sorry I can't be more helpful.
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