Matchweed or Frogfruit
The native plant is called Matchweed or Texas frogfruit, Phyla nodiflora. Notice, I did not label it as a weed. I know it is a weed in your lawn because you do not want it there, but this plant is a native plant commonly found as far north as Pennsylvania and as far west as California. It also grows in other parts of the world such as South America, India and Japan.
It has been considered as an alternative to lawn grass and other ground covers for people who are looking to reduce their water usage in the landscape. Frogfruit is a perennial which means it reproduces not only by seed but also by stolon and lives more than one season. Throughout most of the warm season, it produces a small flower which is attached to a stalk attracting many native pollinators. The nectar is a favorite of several butterfly species (hairstreak) and a larval host plant for the Phaon Crescent, White Peacock, and Common Buckeye butterflies.
Attracting butterflies is one of the reasons it is a favorite of native plant growers. Moist areas with sandy soil are the preferred habitats for frogfruit, so consider checking your irrigation system to be sure the area is not getting too much water. The soil in this area could also drain poorly. However, like other perennials, once the plant is established, it becomes difficult to manage. Since it typically grows in a mat form close to the soil level, hand pulling might be another good option.
Your landscape has St. Augustinegrass therefore you might consider managing seed growth by applying Atrazine in the spring (March) before fertilizer application. Remember, we apply fertilizer (15-0-15) in April. Another application of Atrazine could be done in October for winter weed management. However, it is important to note – this weed is a full-grown adult – management of weeds works best when chemical application occurs on very young and small weeds.