Push Notifications from organizational authoritative news site.
In additions to the current Sharepoint app notification rules https://support.office.com/en-ie/article/When-the-SharePoint-mobile-app-will-notify-you-2113fb91-c660-4034-8d5b-6285f47d127e , add the following.
Push notification from News sites marked as organizational authoritative news site. ( Set-SPOOrgNewsSite )
I would suggest that user following site + this should equal notification so the users have some choice too