Hmmmmmmmmmmmmmmmmmmmmm.....
Even if you kill them off right now the dry ones are going to be a bit of a nuisance for a while. When my wife and i bought a new house three years ago we had by far the worst yard on the entire street - crappy sod over left over building sand that was completely taken over by weeds.
I scalped, spread a load of compost, used the natural ferts and by the end of the season (me moved just as the grass was coming back in Spring) it looked really good but was still 'weedy'. After three years, I have almost no weeds because I keep the grass short until the worst heat of July/Aug and that REALLY thickens it up to the point where other weeds (grass is a weed) can't compete with the grass.
Is that St Aug? if so, it is very aggressive and will choke other things out once it takes off. If the other stuff grows lower than the St Aug then the grass will outcompete it for light and nutrients and it should improve quite a bit.
Assuming that your weed there comes back from seed, the corn gluten meal treatment would help considerably, your grass comes back from the roots and is not affected. CGM is also a nice organic fertilizer with a lot of Nitrogen (protein).
I'd use that and something like Milorganite (Cheap, at lowes and home depot, lots of iron and sulfur which your lawn needs and it GREENS it) and use a mulching mower. You will build up the soil and the grass will get much better pretty quickly.
BTW - if that is St. Aug I believe MSMA would kill it just as it would kill crabgrass.
Final thought: If there is any time when the burr producing weed is actively growing but the grass is still dormant then that is the time to spot treat with vinegar. The more of it you get rid of that way, the less can go to seed (produce those burrs) later.
It takes a little while but think of it as a goal, not a chore
Hope that helps some.