An extra PTH run, or Plated Through Holes run, occurs when a multi layer PCB passes an extra time trough the galvanic plating process. A standard multi layer passes one time through the galvanic plating process. When producing multi layers with blind and buried via holes, it can be required to pass a number of extra times. These extra runs introduce an extra cost and need to be introduced in the order details.
More about the Galvanic plating process in this video: