Answer: If the roots are sending up new shoots, the roots are alive and since mulberry trees grow so quickly, you might want to encourage one of the shoots to become the trunk of a new tree. If so, cut off all the other shoots so the roots can direct energy into the remaining one. Eventually you'll have a new mulberry tree.
If you don't want to keep it, I'd recommend digging it up, roots and all. If you simply plant in the same spot, the mulberry roots will continue to send up sprouts and you'll have a real mess after a while.
Best wishes with your garden!
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