Strangely, if you deliberately remove a widget, you get a persistent pop-up that won’t go away… but generally it’s up to the application to manage.
I like the feature I have on Android, with ‘Notistar’ which keeps a persistent backup of all notifications that have passed the lock screen (i.e. if I read them, or swipe them from the screen I can call them up again in Notistar).
It might be a good idea for a request for this - a persistent backup for notifications, perhaps with the ability to log them all with some advanced settings (e.g. Log All for 1 hours, Log critical for 1 week).