Current Situation: Settings > General scroll to 'Incoming Mails'

You can choose: 'Uncheck 'Show an alert' - to switch off pop up notifications.

'Select 'Show an alert' - to use Thunderbird pop up And if you want to use the system notification : select 'Use the system notification'

If you switch the alerts off then you could just use the sound. Select 'Play a sound' Choose prefered sound.

If you do have alerts enabled and choose to use the system notification. You could try this: Click on 'Customise' Show new mail alert for: reduce to 1 second and click on 'OK'

Then you will get a pop up but it will disappear fast. So not much of a distraction. But if you want to check what arrived then you can click on the 'Notifications' icon - bottom right of Task bar tray area to see the list of notifications.

There is no option in Thunderbird to detect whether Thunderbird window has focus before posting the pop up alert.

Thanks, May I suggest to add such option: "detect whether Thunderbird window has focus before posting the pop up alert"?