I'd agree with Michael. You need to introduce some aeration by spiking. Try to spike as deep as possible.

As you,ve mentioned introducing some sharp sand will help a lot.

It will improve in time, not overnight. So the best thing is to keep off it while it's wet.
Compacting the soil in the already waterlogged areas will negate any treatments you make.

If the above doesn't cut-the-mustard I'd definitely consider improved drainage.