Answer: If the weed is a member of the Solanum family, it's probably one of the nightshades. The berries are poisonous, so be careful around the plant. The secret to is to keep the plants from flowering and producing berries. The plastic mulch you're using eliminates light, which keeps the seeds from sprouting, but anywhere the plastic has an opening sunlight can penetrate and seeds can germinate. You can place additional mulch over the plastic to cover any slits or tears and keep sunlight from penetrating. Since the weeds are annual and should be coming to the end of their lives, pull any remnants from your garden. Then add some mulch material over the plastic. Next spring, if new weeds sprout, pull them while they're small and before they have a chance to flower and set seed. If you keep at it, you'll eventually eliminate the pesky weeds.
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