There is the button Clear History at Advanced - Configuration - Toolbars / Menus - Miscellaneous which clears several histories including the projects history.
Another method is to simply click on the listed projects. If the project file (*.prj) does not exist anymore in the stored directory, you will see the error message "Not a valid project file." and the project is removed from the project history list. So you can temporarily rename the *.prj file of old projects or remove it temporarily to a different directory, click on its name in the project menu to get it removed from the list, and rename or move the project file back.
The last method would be to open in Notepad the INI file of UltraEdit while no instance of UE is running and edit the list under section [Recent Projects].