There is a ring shaped parts between the cover assembly and the pressure plate of the EXEDY multi plate clutches, but what does this do?

This is a part called “pivot ring”, and it will enhance the clutch engaging performance.
In the normal clutches, clutch engaging performance is enhanced by placing a wavy cushion plate between the facings, but this method cannot be used for the metal and carbon materials, so the “pivot ring” is reducing the shock when the clutch is engaged as a disc spring, in turn enhancing the clutch engaging performance.