Compacted, patchy lawn am committed to organic answers, can you help?

Have a small front garden facing South East. I live in NW England where it rains often, currently I have a patchy, compacted lawn, growing in clay bound soil. I am committed to organic gardening so if anyone can offer advice, I would be very grateful, thanks.

I was about to aerate, lift, scatter compost seed & more compost & water but I feel it may be wasted. The grass does not show strong root growth. The ground is quite spongy & even in summer, still a little damp.

 

Thanks Andy, I shall get searching & reading

 

Excellent thank you very much, I hall get stuck in