You can go to Setup>School>Parent/Student Access>Available Features tab. There are three push notifications you can enable/disable there. More often than not, users manage which notifications they get in the Mobile App itself.
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Currently, there is not an audit for notifications sent in the PowerSchool Mobile app. I have submitted a feature request to the PowerSchool Product team on your behalf to enable an audit of notifications.
@JeffG9 Jeff mentioned a great process to determine and manage the notifications that are sent to users.