Chickweed has white flowers. Speedwell (Veronica) has purple flowers. You could have both. Both are winter annuals--and should die out in hot weather. New seed sprouts are particularly hard to spot at first; tiny, they usually hide in the shady areas near the other grass. It is normal for a few seeds to lay ungerminated in the sunniest spots.
Feed to stimulate thickness, and water for now. Plan on weed control in mid August. Then add additional seed, if needed.
Winter annuals like chickweed and speedwell are best sprayed in late October--but it may be too early this year if your plant new seed. (Should be OK if the new seed has been mowed twice.)