Lawn: Scarify it for starters. You may be able to hire a mechanical scarifier to make it easier. Spiking would help too as you get moss when drainage is poor - in this instance because of the clay. Its getting a bit late to do it but there's still time. If you are keen enough you can dribble coarse sand into the holes you make when spiking it...

Improve the rest of the soil by adding anything that will make humus - compost, leafmould, rotted manure. and perhaps some grit or coarse sand as well. This will help to discourage the clay particles from sticking together. If you improve one area at a time you'll see results quicker than if you spread it thinly all over.

Plants for clay: Roses, asters, perennial geraniums, Achillea, Russian sage, daylillies for starters. Hardy fuchsias seem to thrive in it in my garden too - worth a try.

As well as Steragram's plant suggestions...

Euonymus, weigela, pyracantha, hellebores, jasmine, primroses, ajuga. All of these do well in my sticky clay, though I do add composted bark and grit to the planting holes.

I got rid of all the grass-cum-moss (far too much hassle on heavy clay).