Looks like a tree/shrub seedling. Maybe popcorn tree, AKA Chinese tallow tree.

it's not that. It's very soft and never grows taller than 8-9 inches.

Sunflower?

It "never grows taller than 8-9 inches." Is that because of the mowing or pulling? Or have you allowed some to grow and that was as tall as it got?

I allowed some to grow and they only got 8-9 inches tall in one season... then died back in hot summer. Then it starts all over again. I don't think it's a Sunflower. The one in this picture is very small... but left alone, it sprawls and spreads without any flowering that I've noticed.

Clover once was considered a desireable plant and not a "weed" because it can help improve soils. Just scuffing the mulch should be enough to uproot what is growing there.
I have seen people spray things like 2,4,-D near their Lilacs and other plants, with some kind of barrier to keep the spray off those plants, only to have those plants show signs of herbicide poisoning because, apparently, the plants uptake that poison through the roots. Not supposed to happen according to the producers of these products but it does.

I would leave the clover there, but it will then be throughout that entire bed, and that's not the look I was really going for. I wanted to keep it neat and accessible.

It's apparently Chickweed. Just found it.

It doesn't really look like chickweed to me. That is extremely common and I think some would have suggested it before now. 8 - 9 inches is tall for chickweed, it doesn't have red stems and the leaves are less pointed. It also has small white flowers which you don't mention.

I haven't seen it flower, but I have to say I can't guarantee that it doesn't. The 8-9" growth is only against a fence... which I guess if it's trying to sprawl, might explain why it gets high on one side? If I hadn't just hand-pulled so much of it out, I'd take a picture of a "big one."

I don't see any chickweed in your pic.

Hmmm... I have so MUCH of this, and can't seem to get an ID on it.

Look up "horseherb" and tell me what you think.

No, not horseherb. I looked it up and the leaves are too veiny and thick. These leaves are flat and thin and fragile. I will go outside when the sun comes up and look for a picture of it after it's grown larger.

Any progress figuring this one out...?

The company we hired to spray the front yard still says it's chickweed. I just took these pictures of several different "more mature" plants that have been allowed to grow for about six weeks. Let's see if anyone knows what it is now?

{{gwi:144254}}

Huh, doesn't look like popcorn tree anymore. Those leaves and red stems look Impatiens-y to me. Are there little flowers along some of those longer stems? If so, I don't know an Impatiens to suggest, but FL plants are a whole separate area... Might put these on NTP if you don't get an answer here soon.

I'm going to run some errands, then try to remember to put something in the picture for size reference. Maybe some white paper under it so you can see the growth habit. The fifth picture down is one plant.