A good point, thanks GemmaJF
I would not amend the clay soil as it is just grass but if drainage is an issue (sitting water for more than ten minutes after heavy rain) then I would look at the lowest point in the lawn, cutting the turf up, remove all the clay and backfill with grit/rubble, cover the top 6 inches with compost and relay the turf. That should help as a quick and reliable fix.
Inpartially reseeded my lawn which is on heavy clay. I took some clay off it and topped with screened topsoil. However, the old lawn seems to be on a bed of sandy loam and red granite stones (like the decorative ones you can buy). It drains really well but grows well too. Not orthodox but seems to work. Grit seems to be the way to go to help break the soil up.