To prevent users from using drag-and-drop and context menus to alter the Start menu, you can Enable the Disable drag-and-drop context menus on the Start Menu Group Policy at
User Configuration\AdministrativeTemplates\Start Menu & Taskbar

NOTE: This policy will also remove the context menu from the Start menu.

To do this via the registry, use the Registry Editor to navigate to:

HKEY_CURRENT_USER\Software\Microsoft\Windows\CurrentVersion\Policies\Explorer. Edit or add a Value Name of NoChangeStartMenu, a REG_DWORD data type, and set the data value to 1.