My lawn, in certain parts, has been overrun with this weed for years. No sooner do I kill it off, and scarify, it's back, and with a vengeance!

Can anyone tell me what type of weed it is and how I can get rid of it for good?

It is in a part of the garden which gets very little sunlight, is often damp, and where the grass coverage is patchy, at best, despite me using shady grass seed.

Any advice welcome

Cheers