Robinia pseudoacacia, black locust, is rarely seen in residential areas and is not recommended for the home landscape because it tends to be invasive, weedy, suckering, thorny, and has brittle branches that can break off in the wind. If  the pods are ingested, it may cause death in humans and livestock and pets. Depression, weakness, dilated pupils, vomiting, bloody diarrhea, weak pulse, coldness of arms and legs, paleness, and shock.
Suckers are very difficult to control once they start appearing. Since the best known option is to use an herbicide at this time of the year, I would strongly recommend hiring a qualified certified applicator to do this job. This process may have to be repeated over time. Cutting the sprouts can actually increase the sprouting unless done over a period of time as long as any green is showing.