Thank you so much for sharing the photo. I found it in my yard for the first time this year too, but I pulled it, so I have no seed issues. This weed is called annual trampweed, Facelis retusa. The good news it is about to go away because it shows up here in the late fall or early winter.
The bad news? It is an annual and it produces hundreds of seeds. Remember, annuals can only reproduce by seed. At the top of each branch will be a fluffy, white clump of seeds. If you don’t hand pull the weeds or bag them when you cut the lawn, then there will be hundreds more next year. Annual trampweed does have a prostrate growth habit which often makes it difficult to get all the seeds when mowing.
Best preventive method is to keep full, healthy grass. Fertilizer using 15-0-15 on April 15 and September 15. High nitrogen fertilizers can cause disease and insect issues. Water in the early morning hours, never late in the afternoon or evening. St. Augustinegrass likes to be watered deeply, but less often. The goal is ½ – ¾ inch of water each time the lawn is irrigated. Make sure the grass is getting the same amount of water each time and in all areas.
Avoid adding lime unless a soil analysis from UF/IFAS recommends it. We recommend pre-emergent herbicides and for this weed, the application should be done in October.