I tried the other day to unpin the Public Desktop folder from the Jump List of Windows Explorer in the taskbar. Mission Impossible, and it was, the item refused to disappear! I had a similar issue before, on Windows 8. If you face a similar problem, press Win+R to go to

%APPDATA%\Microsoft\Windows\Recent\AutomaticDestinations

Press F to find the item f01b4d95cf55d32a. Delete it, and ALL the items in Explorer’s Jump List will be removed. Of course, you will lose also items you’d like to have there, but the list will rebuild itself over time as you open and close your folders.

Note also that you can achieve the same result by going to Settings > Start and deactivating the option to “show recently opened items in Jump Lists”. Deactivating this option clears instantly ALL your jump lists not only that of Windows Explorer. Even if you activate it again immediately, your jump lists remain empty and are being rebuilt over time. This method is the one you should follow if you’d like to make a new start with all of your lists.

Enjoy!