@mh
Here's the list of hacks I've personally collected (I'm guessing you're talking about iOS.)
The easiest is to just have a notification feed on a website (possibly with an RSS feed and maybe even a mailing list.)
Most OSes give the built in mail client special privileges that allow it to slowly poll (but not get pushed to via the imap IDLE verb) imap. Interestingly the iOS notepad app gets update from another mail folder (you can edit it in mutt to) which can give you a funky delay tollerant wiki. I have a program I wrote a while ago that will watch mailboxes for updates and push the notes to a static html directory (the annoying part is that pictures are inlined as bas64 by the notepad app which is a pretty awful way to handle that but idk what else you would do with that architecture.)
Most OSes have a CALDAV client that will poll for updates. But then you have to run a caldav server which sucks.
Many messaging apps allow users to create custom webhooks to trigger push notifications. Discord supports this. Colloquy has a push relay server intended to be used for notifications from a remote server (usually a bouncer but I have a fork of miniircd somewhere that lets you see mentions in channels you've been in as push notifications.) Unfortunately there's just one person maintaining it so the availability wasn't great last time I checked.
iSH can run in the background but it requires turning on the GPS front end amplifier (from what I understand) so that will drain your battery very very fast.