You didn't mention what kind of tree these grow around, but they look very much like root-sprouts from the tree itself. A number of trees will do this.

Compare the leaves of the 'weeds' to the leaves of the tree. They may not look exactly the same, but are they very similar in the details?

If so, there isn't a lot you can do to 'fix' this. But the mulch circle you have around the tree is very small. You could dig down some in this small area, put down heavy plastic, and then mulch this small circle more heavily. Won't interfere with your tree's growth because the area is so small. The roots involved in uptake will be beyond this circle.

The other ones, in the area that is being mowed anyways, just ignore them. Mow them down as you mow the grass.

Linda

We aren't sure exactly what kind of tree it is but my wife thinks it's a pear tree. How do root sprouts grown? Is this from something dropping off the leaves or something to do with the tree roots?

They are sort of like suckers from a lilac; cut them off just under the ground. New ones will probably keep coming, though.

Here is a link that might be useful: about root sprouts

The tree roots just keep developing new above-ground 'sprouts'. Happens with a number of ornamental trees, especially ornamental fruit trees.

In the morning, compare the leaves. If possible, send a photo of the tree's leaves. I really think you are dealing with root sprouts here, not weeds.

If they were weeds, they would be in other areas of the yard as well. If they only keep recurring under and around this tree.... Then, yep.