hi I'm new here and gave same problem, you need to scarify your lawn either with spring tined rake or hire one, put down a moss killer and reseed, I used a lawn feed last year with moss killer in it, it done a great job, but I didn't do quite as good a job, didn't scarify properly or reseed. Add top dressing enough. I'm on my second attempt at salvaging my lawn, can't afford to returd so breaking my back doing it the hard way!!! Good luck :) Mags

Its a little late to scarify - wait till autumn to do that. If, though, the lawn is very wet, then you can spike it now by inserting a garden fork into the ground about every 9 inches right across the lawn.
In the meantime, buy yourself a lawn weed, feed and mosskiller treatment, apply according to the instructions. When the moss turns black, rake it out and if you then have bald patches, prepare the areas and reseed, but don't attempt to seed until 6 weeks has elapsed from the time you applied the treatment.