Me here wrote:
It sounds like you'll need to lift all the turf (Not all at once though) and
remove a fair depth of the clay and replace with a quality soil or loam. Add
sharp sand at the same time to give a good drainage layer, re seed and lay
the turf back over.

Yep, just dig it out, replace with decent stuff and re-seed or turf.
You could put a layer of compost sand mix over the top of the clay
which will help a little as a blotting paper arrangement and will stop
the clay from drying so quickly.