I would make sure that the area is getting enough sun, and maybe prune up and/or window nearby trees--at the proper time. Improving the drainage with aeration, and limestone or gypsum, depending on the pH.

I agree the lack of drainage is the problem but it wont be helped if the lawn is shady. better aeration needed.
spike the lawn and brush in sharp sand to help with the drainage. Mow the grass so it is a little longer too.

Scarifying with a lawn rake will remove some and it would seem that Sulphate of Iron (as used for moss) will slow its growth. But as already advised it is a light and drainage problem which will be improved by spiking and the addition of sharp sand.