A stop at a nursery last evening elicited the answer: what I am try to eradicate is mallow. The nursery has a spray that will kill it quite dead -- and also the surrounding flowers, grass, shrubs, and vegetables.

Shucks, I've got some extended release Round-up that'll do that AND make the ground fallow for three months thereafter. (I use it on the golf course fence line and on the stuff that comes up in the cracks in the driveway. You can also use it to write stuff on your neighbor's lawn if you don't like your neighbor, but I wouldn't know about things like that.)