In Windows 10, you can use the Windows key + V shortcut to bring up your clipboard history. If you don't have clipboard history enabled through settings, you will see a button to Turn On the history. Once it's on, you'll see the Pro items listed. You can select them individually to delete or clear all items at once. Other than that, I usually see the pop up you posted when I exit a project in Pro and clear the clipboard that way. I haven't found a way within Pro other than clearing on exiting a project.
