Hello, 

1. Pull seedlings when the soil is damp and get the crown/root out.
2. Re-mulch immediately and keep mulch at 2–3 inches.
3. Never let seedings mature and set seed.
4. Trash pulled plants instead of composting them.
5. If the infestation is severe, consider a pre-emergent in spring or careful spot treatment of existing seedlings.

Here is a link to controlling unwanted Mulberries from University of Illinois extension.

https://hyg.ipm.illinois.edu/article.php?id=1313

Some of the information I found suggests it could be mulberry weed but the treatment is the same as if it was a seedling.  https://extension.psu.edu/mulberry-weed

Here is another response from a master gardener.

Removing numerous saplings can be a daunting task. If you don’t want to dig or pull them out then it is easiest to use some type of chemical control. You will need to purchase a product that contains 8 to 10% glyphosate or triclopyr. You will also need a small paint brush. I like the foam brushes, but any small brush will do. Cut the sapling near the ground level and immediately paint the cut with the undiluted chemical. If you have someone to help you, it is most efficient to have one person doing the cutting and the other following right behind with the chemical. You don’t want to cut all of the saplings first and then apply the chemical, because the saplings will begin to seal off the wound shortly after being cut, and the product will not be effective. You must be very careful while doing this procedure. These chemicals will kill desirable plant material, so do not accidently put any chemical on anything that you don’t want to kill.  Once you have completed applying the chemical, dispose of the paint brush. The following websites will provide more detailed information about eradicating undesirable saplings. When using any chemical, be sure to read and follow all of the directions on the product label.

http://edis.ifas.ufl.edu/ag259

www.uaex.edu/publications/pdf/FSA-6124.pdf

Once you have eliminated the current crop of seedlings watch for any new sprouts. They are much easier to pull out when they are very small.

I hope this information is helpful. 

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