Corn speedwell (Veronica arvensis L.) is a winter annual that produces vegetative growth and flowers from early spring through summer. This weed reproduces via seeds, which germinate during cool, moist conditions in late summer, fall, or spring.  Broadleaf (post emergent) weedkillers such as 2,4-D, 2,4-DP and dicamba, or 2,4-D, clopyralid and dicamba are effective but apply before it goes to seed.  Pre-emergent weedkillers are also effective but must be applied by late summer to stop seed germination.  A good defense against corn speedwell is dense turf from an extra application of nitrogen and irrigation.
More info on corn speedwell and it's control in the following articles:
https://www.canr.msu.edu/news/spring-blooming-lawn-and-garden-weeds
https://extension.psu.edu/lawn-and-turfgrass-weeds-corn-speedwell
https://ag.umass.edu/turf/fact-sheets/speedwells-in-lawns
I am also going to suggest a soil test if one hasn't been done in the past 2 years.  You will learn more about your soil including soil type, pH, nutrient levels and more. Knowing the type and fertility of your soil is the starting point to establishing & maintaining a strong turf that can resist diseases, weeds, moss, insects/grubs and look great at an optimal cost. More info here: Don't Guess-Soil Test-MSU and MSU Home Soil Test-self mailer