"Relocated items"

If you had items stored on the root of the file system, then those items were moved to the "Relocated Items" folder in the "/Users/Shared" folder. You can move those items from that folder to somewhere within your home user folder. See this Apple article for more details:

About the read-only system volume in macOS Catalina - Apple Support

It usually depends on the app. Some apps allow you to configure a default location, while other apps tend to use the last location used, while others allow you to select the location. Do you have an option to "Save As...." which allows you to change the location (you may need to hold the Option key while clicking on the File menu for the app). Maybe this will change the default location for the next new save. Check the app's Preferences to see if it has an option to configure the default location for saving files (some apps don't have this option). If the app does not have such a configuration option, then you could try deleting the .plist preference file for the app (maybe the preference file holds that location somehow even if it is not a configurable option).

Is there one specific app with this issue, several apps, or all apps? I'm not a macOS software expert, so maybe another more knowledgeable contributor will have some ideas or be able to assist with a specific app(s).

Why does a Relocated Items folder appear as the default destination when I try to save to my desktop after specifically selecting desktop as the destination? In order to access my desktop from this folder, I have to click on another destination and then click on desktop again.