On iOS you can't do this, as you've described it. You don't get to schedule tasks to happen at regular intervals when your app is in the background or not running. You also don't get to control whether iOS decides to terminate your app when it's running in the background, so "not running" is a case you'd need to handle (you can't just keep running in the background as for long as you want).
I'm not sure I understand the server side of things though. If your server is manipulating the data until it's acceptable, and it can send push notifications, why does it need to wait for an incoming request from the phone? Why not just send the push when the data is ready? If the app decides what's "acceptable", maybe have the app tell the server what it wants so that the server knows when to send a push.
There are a couple of options that would get close to what you describe. If you implement the "background fetch" feature in your app, iOS will launch the app when it's not running and let it make network calls in the background. There's no guarantee of how often this happens, though. This is described in Apple's background execution docs
The other option is the "silent" push notification. If your server sends one of these, iOS can launch the app in the background to handle the notification. The app could make a network call if necessary. You can send these at whatever time you like, but Apple warns to not overdo it:
Silent notifications are not meant as a way to keep your app awake in the background, nor are they meant for high priority updates. APNs treats silent notifications as low priority and may throttle their delivery altogether if the total number becomes excessive. The actual limits are dynamic and can change based on conditions, but try not to send more than a few notifications per hour.
Silent pushes are described in Apple's push notification docs.
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