This is my first time dealing with poa annua. Also in Missouri with fescue. I've been pulling up the obvious spots and bag mowing every 3-4 days. Luckily, it's only sparse in my recent lawn reno area. Did you reseed or reno last fall? It is interesting to see how prevalent poa a and t have been this spring.
I plan to pre em the reno area this fall and spring. The few areas that need reseeding will instead get Scotts Starter Fertilizer w/Mesotrione, Tenacity 30 days post germination or after a couple mowings, then wait another month to put down a pre em. Pull up whatever is white and doesn't look like baby fescue. I have seen this suggestion on the forums but am wondering how the 2 month old grass would handle the pre em.
There is also Poa Constrictor, which seems to have mixed results here and the even more expensive Xonerate. You could always kill, fallow, reseed, etc.
Though, I care more about time vs cost. If poa annua becomes a problem down the line, I'll just buy Xonerate and do a 4-6 way split. From what I've read, it seems to be fairly effective.