Floral, I didn't even know there was more than one Aphanes species in the US. The following Weakley snip suggests that A. arvensis has larger leaves. The BONAP distribution maps suggest that the Georgia plants would be A. australis.

Shotweed, Cardamine hirsuta, aka hairy bittercress (or a vey closely related species). A cool season annual that can be easily pulled in moist soils. Any broadleaved lawn weed herbicide will work but try to pull them or kill them before they flower and set seed (which yours have already done) as it has a large seed bank and as the common name implies propels ripe seeds over a large distance.

If it comes back that rapidly, you have either not removed thoroughly or the weeds have aready banked a lot of seeds. Spraying for weeds in an established planting bed is just not a highly advised practice becasue of the strong potential of herbicide damage to any existing desirable plants. Make sure you get any deep roots out with the topgrowth of the weeds. And a 2-3" thick layer of mulch (without any weed block fabric!!) will stop the weed seeds from germinating. I see no evidence of any recent mulching in the bed in the photo....... And despite whatever method of weed removal you may follow, it will never be permanent. Weeding is a routine part of gardening :-)

Appears you have Annual Poa. Very aggressive and produces heavy thatch that crowds out good grass types. You can applie an pre-emergent to prevent the seed from the seed heads from germinating. Best I found was to De-thatch your lawn and use a crabgrass killer and pre-emergent to remove it. Next opportunity you have to re-seed, Fall or Spring (make sure you wait for the pre-emergent and crabgrass killer has dissipated ~ 3 to 6 months) then re-seed.