I had the same problem in my garden, and honestly, the turning point wasn't some single measure, but a combination of things. The most important thing I realized: moss is actually a sign that the lawn itself is too weak and can't compete. If you only fight the moss without strengthening the lawn, you get exactly what you're describing - it just comes back.
What helped me was seeding the bare patches deliberately after scarifying and then fertilizing regularly (especially in autumn and spring). It might sound like a lot of work, but a dense, healthy lawn just smothers the moss on its own. At the same time, I tried to improve drainage in the wet corners - not with major renovations, just by improving aeration and letting water drain away faster. Some people swear by moss killers with iron sulfate, but in my experience that's just a quick fix that kills the moss without solving the real problem.
The key thing is probably this: look at why the moss is growing there in the first place, not how to get rid of it as fast as possible. Too much shade? Waterlogging? Poor soil? You have to tackle it depending on what's causing it. For me, the problem only really went away when I started making the lawn itself healthier instead of just ripping out the moss.