How to stop messages appearing after MacBook updates

If you are referring to that specific process, AddressBookSourceSyncScheduleHelper is the process responsible for managing and timing the synchronization of your Contacts app data. It works in tandem with the primary background daemon AddressBookSourceSync, ensuring that your local Mac address book stays updated with remote cloud services like iCloud, Google, or Exchange.

If you mean more general notifications about apps running in the background, that's normal for current macOS security policy to inform you of processes you may or may not want running. For example, Adobe Photoshop wants to run in the background on my Mac, and I see no need for that.

To stop macOS "Background Items Added" or persistent background activity notifications, manage your login items.

Manage Background Items

• Open System Settings from the Apple menu.
• Click General in the sidebar, then select Login Items & Extensions.
• Look under Background App Activity and toggle off any apps or developer helpers (like Google or Microsoft update tools) that trigger recurring alert

More info from Apple: Manage login items and background tasks on Mac – Apple Support

Thanks for the very thorough explanation. I don't want to disable this process, as my contacts do in fact reside on a cloud service. However, looking at it under the setting shown in your screenshot, it says that it's *not* running in the background, even though I haven't disabled it and even after starting the Contacts app. It's touching that Apple is so concerned about my security. ;-) I'm just grateful that the notification only comes after updates, and doesn't come for *all* apps that are running in the background, e.g. apps that support my non-Apple mouse.