I must say that I and many other gardeners in the northwest share your pain. Pearlwort, Sagina procumbens, is a particularly vexing weed that we battle nearly continuously through the year.
This nasty stuff spreads vigorously by rooting at the stems and by the multitude of extremely tiny seeds. By the time pearlwort is in flower – as are the specimens in your image – it’s also producing seeds! What may be even more aggravating is that pearlwort sets seed and sprouts from April through September!
It would be great to win the Weed War. Instead, expect that the weedy siege will continue as long as you garden. You can decrease the pearlwort population if you persist in removing every seedling as soon as you see it and definitely before it sets seed.
Glyphosate, the active ingredient in various home garden herbicides such as Roundup, will kill pearlwort. “Painting” the product of choice onto the weed eliminates the need to spray and will protect nearby plants. Just as with hand removal, you must act before the plants set seed, preferably before they even bloom.
I hope these insights will help you plan your attack,