How do you rebuild the desktop on Catalina 10.15.5?

I understand you had items on your Desktop, but they vanished. Is it possible they are on iCloud Drive?

Refer to this excerpt from Optimize storage space on your Mac:

Recommendation: Store in iCloud

Description: Helps you store files from your Desktop and Documents folders in iCloud Drive, store photos and videos in iCloud Photo Library, store messages and attachments in iCloud, and optimize storage by keeping only recently opened files on your Mac when space is needed.

See Use iCloud Drive to store documents on your Mac, iOS, and iPadOS devices, Use iCloud Photos to store photos in iCloud, and Use Messages in iCloud.

You can modify these settings later in iCloud preferences, Photos preferences, and Messages preferences.

If that doesn't address the mystery then I have no explanation. Try searching for them using Spotlight, but if they can't be located or were deleted for some reason, the usual way to recover is to restore those items from a Time Machine backup.

Describe what it is you would like to accomplish.