Watersprouts or suckers are a big problem with Chinese tallow (Sapium) and lots of other trees. When you water, water deeply so that you can be sure the tree isn't sending up suckers to get more water. All I do is cut my tree suckers to the ground and mulch around the base, making sure I cut the suckers back again as they reappear. You could use napthaleneacetate, the active ingredient in Sucker Stoppper RTU, if you want a more drastic measure. It doesn't appear that you want to use the strong stuff, so try to live with just pruning and mulching.