I don't know what it is, but have you tried a regular broadleaf herbicide on it? I see some crabgrass and nutsedge in there too, so you can try Roundup for Lawns on all 3.

I applied Scott's Bonus S ten days ago. It's probably too early to spray Roundup to the affected areas.

Note: It looks like this weed is burweed....

Oh be careful with Roundup. Regular Roundup is glyphosate and it will kill everything. Roundup for Lawns (green label) is a mix of selective herbicides safe to use on cool season grasses. It has dicamba, sulfentrazone and quinclorac I believe. They make one for southern lawns too (Bermuda etc)

The branding is very unfortunate, someone is bound to get the wrong one sooner or later.

"...Prostrate spurge is a common lawn weed. A low growing, mat-forming summer annual that thrives in hot, sunny and dry sites..."

It isn't prostrate spurge. OP says it's burweed and that looks like a fit.