clay

If your Clay Layer is that close, you will be best to Try to Till through it and add Compost and lime possibly.
Or consider going to raised beds where you can control your soil condition

Another way you could try is to add a lot of course sand with peat – it will take time and will brake down but is time consuming - plus planting crops to brake down the clay with there roots – but it is hard work and time consuming to - It would probably be better as suggested above raise the bed slightly rotivate what you have add sand as I suggested and then add fresh top soil – I did this about 20 years ago and now I have good soil. It all takes time

I had inherited a similar garden. Heavy clay.

I first prepared the bed area, dug out about 8" of the clay soil and broke it up as much as possible. Then, every year piled in my home made compost and some sharp sand for drainage. Took years!! But now I have beds with a foot deep good gardening soil.

Make your own compost. Compost all your fruit and vegetable peels, egg shells, garden waste at the end of the growing season, grass cuttings and throw in the brown cardboard rolls from toilet rolls. Its a very economical way of creating useful compost and also very eco friendly.

Good Luck.