The iCloud files are cached on your Mac, so you can access them even when you have no internet connection. They are stored in ~/Library/Mobile Documents/.
To make it even more accessible, you can add the folder in your sidebar.
There you'll see that each application has it's own folder with it's own documents. Files can be added or removed in this window.
To make sure you're working in your iCloud, you can try and delete a file. Your Mac will notify you, you're deleting the file in iCloud.
Doesn't look like there's a way to single out the Desktop via the 'Manage Storage' GUI.
However, you can always just symlink any directory from your iCloud drive; that way you can ensure that only the Desktop folder is being synced. This is with the caveat that while your Desktop files are being stored in iCloud, they're still taking up space on your HD before, during, and after sync. If you do it on every Mac then you'll keep your Desktop in sync between them all (files added, deleted, changed etc); this might not be what you're after though.
If symlinking isn't something you know how to do, instructions are below.
Backup (via TimeMachine or whatever you use).
Then fire up Terminal and run the following to move your Desktop folder into your iCloud Drive:
mv ~/Desktop /Users/<yourusername>/Library/Mobile\ Documents/com~apple~CloudDocs/
Then just symlink it to the usual 'Desktop' location:
ln -s /Users/<yourusername>/Library/Mobile\ Documents/com~apple~CloudDocs/Desktop /Users/<yourusername>/Desktop
Now anything that you add to ~/Desktop (by dropping things onto your desktop etc) will actually be dropped into your iCloud drive. They are of course still taking up local space on your machine. When you delete something from your desktop then it's gone from the 'Cloud' as well. The benefits of this are more for sync between machines (+ access via your iPhone of course).
If you've got another mac, move anything you value from it's desktop folder into your iCloud Drive 'Desktop' folder first, then delete ~/Desktop on that machine and symlink again as above.
Best Answer
It is possible to manage this - basically you have two options:
1) Let macOS handle this automatically
You enable this from System Preferences by clicking Apple ID, iCloud and then ensure that Optimise Mac Storage is checked.
This ensures that everything is stored in iCloud Drive, and as much as possible is also stored on the Mac. However as available space becomes low, the system might delete files from the Mac so that they're only stored in iCloud.
In this case you have no control over which files and folders macOS keeps on the computer and which are only available in iCloud.
2) Handle it manually yourself
In the Finder open iCloud and right-click the folder and select Remove Download. This removes the folder's contents on your Mac, but keeps it in iCloud Drive.