Some good suggestions above..
I found applying a spring grass feed a couple of weeks before applying any moss killer,
we used granular in both cases,
sadly folk with no knowledge ordering the same thing year in year out and not willing to listen.
Moss can be anything, cutting too short, not cutting, too wet too dry and lots of other reasons.
Once you've scarified (raked)(machine or rake) it out,
if the grass can stand up to it, cut, cut, cut on a very regular basis.
Grass grows up the way, the more you cut the more it will spread sideways.
A Summer feed around the beginning of July and more of the same..
End of grass late autumn, feed with an autumn winter type feed then scarify (no mosskiller) which removes any thatch not picked up when cutting , I'd try lighly first and if you can pick up a scarifier(good springbok rake will do) the better but as I said lightly then repeat the above the following year.
Hopefully all goes well....