Look like seedlings of Albizia julbrissen(commonly called 'mimosa'), or those of sensitive plant, Mimosa pudica - though there are some small-growing groundcover-like plants that have similar leaves.
If you can carefully spray with glyphosate - making sure not to get any on the blackberry plants, it'll take 'em out. Otherwise, you're likely relegated to pulling them by hand.

I can't imagine those are much competition for blackberries.

They look attractive and low growing. Is the reason for getting rid of them aesthetic or practical?

aesthetic. I planted the blackberries near my back property line in my pine needle island, not a garden, I guess I should put some weed barrier down around my plants and I can spray the rest of the island again. I just didn't now if some kind of spray is out there like Weed and Feed but for berry plants.

Not that I'm aware of... a broadleaf herbicide or non-selective, like Roundup, would be indicated. But,even with the 'low-volatility' 2,4-D products, you might get some damage due to off-gassing.
Roundup in a hand-spray bottle - with good control, and no wind - would be my choice, if I weren't willing to pull those little babies by hand.

I would cover with a tarp or plywood and smother them

I have those same weeds and so far they have proven immune to all sprays. The only thing that works is to hand pull them - easiest when they are small like in the photo.

Soak a rag with round up and lightly run it over the weeds. That way you don't have to worry about overpay.

This post was edited by insteng on Sun, Jul 13, 14 at 21:24

Compare your weed leaves to spurge. Spurge is a very rampant self-seeding weed. Some were allowed to flower and seed in your berry patch. Perhaps a hoe or other hand tool would be faster than pulling.

Thanks for all the replies. Since there seems to be no spray that wont kill the blackberries I've decided to rake up my pine needles, pull as many of the weeds up, put down a weed barrier silt just around the blackberries and rake the pine needles back. I will continue to spray the weeds that wont have run-off toward my berry plants.

I think the weed is Chamber Bitter, Phyllanthus urinaria.

Are there thorns? My first thought was Devil's Thorn (Tribulus terrestris).

No thorns, I've decided to pull them and put down a weed barrier fabric. Thanks.

I'm pretty sure it is Creeping Charlie. Can't get rid of it, only control by pulling at least once a month. Won't hurt berries, but it is a mess, spreads to the lawn, and hides snakes. I hate it.