How can I stop these old messages from popping up in the top right corner please?

For everything but Software Updates, go to System Preferences > Notifications and turn off notifications for the affected apps:

For system update notifications, open System Preferences > Software Update, click the "Advanced..." button, and deselect

Check for Updates" in the resulting window:

I don't know what system you're running but on Big Sur make sure every app listed in the System/Notifications preference pane have Notifications turned off: