Are you sure it is just clay? Sandy clay packs even tighter than just clay: https://www.uaex.uada.edu/farm-ranch/crops-commercial-hortic...

"If you mix sand into clay, the clay particles will fill in all the open spaces between the sand particles and often the clay will act as a ‘glue’ sticking all particles together, ultimately resulting in a more dense soil."

When it's wet, but not saturated - like 1-2 days after a rain - you can decompact the soil with a strong metal broadfork and leave the soil in large block aggregates. This keeps the soil structure and maintains some fungal web connections. Add nutrients, wood chips, stick and sand below aggregates and in cracks. Cover with compost and plant clover to cover.