It's a little hard to tell from these photos, but it could be smartweed or prostrate knotweed Polygonum aviculare. Do you think this could be the weed that you have?

https://crops.extension.iastate.edu/encyclopedia/prostrate-knotweed

https://www.canr.msu.edu/ipm/diseases/prostrate-knotweed?language_id=

If this is your weed, it is an annual species that is an indicator of soil compaction. Doing things to alleviate further compaction, aerating the soil, and increasing organic matter by leaving grass clippings or mulching your leaves into the turf this fall will help to strengthen your lawn. You could also consider doing a soil test to be sure that there isn't an imbalance of nutrients in the soil. 

As far as managing the plants that are there, you are doing the right thing by hand-pulling to prevent seeding. Herbicides can also be used to control prostrate knotweed, however they will not be effective this late in the season with how mature your plants are. Commonly used chemicals include 2,4-D, oxyfluorfen, clopyralid. For best control, spray herbicides at the seedling stage (next spring/summer). As the plant becomes more mature, it will become more resistant to herbicides. The plant is moderately resistant to 2,4-D even at the seedling stage, and it should be combined with another herbicide (most safe-for lawn broadleaf weed-killers use multiple active ingredients. Preemergent herbicides can also be effective in lawns and perennial landscapes. Always read and follow the entire label when applying herbicide.

Please reference these resources about soil testing, avoiding herbicide misuse, and dealing with compacted soils.

https://homesoiltest.msu.edu/

https://www.canr.msu.edu/news/avoid-herbicide-misuse-at-home-things-to-know-before-you-buy

https://www.canr.msu.edu/resources/what_to_do_about_compacted_soil