I only want a desktop space on my primary monitor (Macbook Pro), with only full screen apps used on my second monitor.
Is there any way to disable the desktop space completely on a second monitor?
Apple does not seem to have provided a way to remove the 'normal' Desktop display in Mission Control and only allow Full Screen apps. However, you can broadly achieve most of what you want:
Tick "Displays have Separate Spaces" in System Preferences > Mission Control, then you can have Full Screen apps in a second display.
Then configure applications to only appear in Display 2 (right-click on the app's Dock icon; Options > "Assign to Desktop on Display 2").
While the apps are in Full Screen mode, you won't see the Desktop.