I can't really visualise what weed you are talking about. Could you add a picture please?

Creeping buttercup maybe? :o) but as Cammomile said, a picture would help :o)

Hi, welcome to GoY, the first thing that came to mind was creeping buttercup, but a photo would conirm or disprove, Derek.

Clover creeps about too. Look at the leaves - if they are small and grouped together in threes at the end of the stem it will be clover. if the leaves are larger and have jagged edges it will be buttercups.
Yu can put Weed and Feed on the lawn or use Weedol (but be prepared for the Weedol to make the grass die back somewhat - it soon recovers.

It could well be ground ivy - glechoma hederacea. I have that in my borders and it seems to be drawn irresistibly to the lawn. It grows on long stems which you can pull at and trace back for inches and feet. It roots every few inches, so yes - it does seem to stick to the ground, The leaves are roughly circular with irregular edges. If it flowers, it produces small blue flowers. I sometimes pull it up and put it at the edges of hanging baskets - it makes quite an attractive, trailing foliage plant. I'm afraid I can't offer any way of getting rid of it. I live with it, pulling it up out of the borders as I find it - which is pretty much daily, I'm afraid!