I think you might be referring to a Crepe Myrtle. The sprouts are sometimes caused by stress on the tree. Has it been injured (excessive pruning) lately or has the environment changed ? (i.e too much water, not enough).
Either way, no good solution. Weekly cutting, or when you mow your lawn should suffice, daily seems excessive. Best not to spray any chemical on the suckers(to try and kill them) as they are still attached to the mother plant.
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