The plasma is discarded as there are no cells in it. Whilst plasma contains some useful proteins the volume of plasma from an average cord blood harvest is not a therapeutically relevant or useful volume.
Many plasma-derived proteins previously used to treat disease now have non blood product derived, manufactured alternatives e.g. the clotting factors 8 and 9 are now never obtained from plasma when used to treat haemophilia A and B. Some cord blood banks store cord blood samples whole , however in order to utilise any potential benefit from stored plasma the entire sample would need to be defrosted and processed thereby destroying the more valuable stem cells.