Yes, and it has been solved long time ago already and is now widely supported by both servers and clients.
> I don't follow xmpp/jabber, but I'd hope there's already an XEP for this.
See XEP-0198 from 2004 and XEP-0357 from 2015.
Taking a quick look, this one is marked deferred; I don't know what that means, but it sounds like it's not an accepted standard?. It requires the application push server to be an XMPP server (which I suspect is challenging for client developers, compared to having a https url that takes a POST or similar) and doesn't include a method for a client to request a test push to validate the setup or discuss feedback to the client for push failure.
This doesn't feel solved to me.
I didn't review XEP-0198 closely, based on perhipheral ecosystem interaction, I do think that one is solved.
The client phone app tells your XMPP server which gateway to use (or maybe it's vice-versa, the gateway contacts your server. I forget). In any event XMPP servers like Prosody support this out-of-the-box.