It might be creeping speedwell. (The flowers in the photo appear white)

try pratia.

I thunked Pratia pedunculata too.

It looks like Pratia. It won't kill the grass but it will spread (indefinitely) by underground rhizomes. If you don't want it you will probably have to resort to chemical means - digging it out doesn't work as you can't guarantee getting out every bit of root.

Personally I think grass looks better for having a high proportion of flowering plants mixed in.