Ask Mr. Smarty Plants

We began by trying to determine which mulberry you had. Morus rubra (red mulberry) is native to your part of Delaware; however, we kept finding references to mulberries as invasive, pushing other plants out of their native habitats. Turns out there is a "common" mulberry or white mulberry, Morus alba, which is native to China. Apparently these often hybridize with the red mulberry, and can, indeed, be invasive. 

What we needed to know next was whether or not the mulberry was allelopathic; that is, producing chemical materials which caused the inhibition of growth of competing plants beneath the tree. We found no reference to allelopathy in mulberries; in fact, we learned that the poster child for allelopathy, Juglans nigra (black walnut) can be especially damaging to mulberry trees. 

Bottom line, although we don't know if your tree is native or not, we don't think it is emitting allelopathic substances and killing the plants beneath it. It may, however, be inhibiting plant growth with its very shallow roots, or the shade it provides. All mulberry trees also drop a lot of litter, seeds, twigs, leaves-which may be cutting back on the plants trying to survive beneath the tree.

The mulberry is not considered a very desirable tree, and the non-native is certainly undesirable, as it can become invasive. You will probably have to make a decision about whether the plant material on the ground or the tree are more important to you and take the other one out.