The area right around the sprinkler always stays wettest. Normally after the sprinkler shuts off, it dribbles a bit, and as it runs, it dribbles a bit. As it comes on, the sprinkler begins by releasing an amount of water over a smaller area and then the spray expands to the "normal" size. Same thing when turning sprinklers off. This is why the area immediately around it gets a little wetter. Probably normal but you could run the sprinklers and observe for leaks.

If it is really wet (puddle that remains quite a while or does not go away) there may be a leak or a problem with the sprinkler head.