Thanks Helen,
We are on Moodle 2.9 for the site in question.
In the documentation for 2.9 I found:
Push notifications
Push notifications are all notifications related to the messaging system in Moodle, such as notifications of instant messages, new forum posts, submitted assignments etc.
Push notifications need to be enabled by a site administrator and configured by users in their messaging profile settings (Administration > My profile settings > Messaging).
On the frontpage when logged in as a student the admin block does not appear. On courses it does appear but only contains a grades link.
I will, between other things, continue to search for this.
regards, Paul.