Well, there are many questions in what you ask, which may cause some problems:
My application does not work on user phone.
A tick (without warning) push notification is expected.
If the user has turned off the background for your application, your application will not even be notified silently. Nothing will happen.
It retrieves data and, based on certain conditions, an event is triggered.
Do not forget that you have only 30 seconds to complete any requests. If your request takes too long, iOS will kill the process.
Update (do not run) the Mail application so that it can receive its own push notifications themselves.
It’s simply impossible at all, perhaps with a hacked device: D Apple does not allow many features with its native applications. You can open the mail application, open a new letter for sending, but that’s almost all you can do.
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