It looks like my Galium boreale. The leaves are rounder like that when they're young. Galium species all look similar. Im pretty sure it's a Galium. Do you have Cleavers in your area? They stick to clothes.

They stick to clothes.

They also stick to fur. It took me a while to figure out where the small burs on the cats' fur was coming from.

In my garden that would be Galium apirine, a fast growing annual and common weed of cultivation.

Hmmm.... I looked it up on DDGo and here are the photos they're offering....

and I don't think I've seen any plants like that in the yard.....

None of those pictures shows G apirine. It's a widespread annual weed and I think your plants are more likely to be that than G boreale which is a perennial.

This is the photo that came up from Gallium apirine...that does look a lot like it, before it elongates. So if it is an annual, it will be dead over winter?

Yes, that's the one. It can grow a couple of feet in a season. One of the earliest plants to show in spring from the seed it dropped in summer. Edible if you're interested.

So will this interfere with replanting this bed? Should I be trying to eliminate it? Or just remove it when it is in the way? Should you try to get it out before it gets any larger?

Galium is like henbit and chickweed, in that they all germinate and put on a small amount of growth in the autumn before winter, and then grow full throttle come spring. It's seeds are in the soil bank. You can keep removing them, but you will have to remove the seedlings next year. They are a slight inconvenience that isn't worth losing sleep over. Be cautious removing them, because they can be stuck to plants that you like, and you could possibly yank out a good seedling with the Cleavers. The leaves on young seedlings are wider and rounder than the leaves of mature specimens.

Young Galium boreale leaves

Mature Galium boreale leaves.

I would remove as many as you can. G apirine can smother delicate plants with its long lax stems.

Will mulching keep them from germinating?

I would get rid of it all. If that's a cleaver, you do not want it going to seed and sticking all over your clothing.