You can get Mixpanel to send your push notifications automatically using the "Notifications" feature- you don't have to log in to Mixpanel every time. You can set up the notifications to be sent based on a user's properties. If you want to send a notification based on some event in your app, you can send a people analytics update to Mixpanel when that event occurs.
So suppose you want people to receive a push notification to their mobile device every time they sign up in your web app. On the "Thanks for signing up" page of your app, you'd put something like the following:
// Assuming you've added the Mixpanel JS snippet to your page
mixpanel.identify(SOME USER DISTINCT ID);
mixpanel.people.set("Signed Up", true);
Then, you could set up a notification in Mixpanel with "Signed Up Is True" in your Mixpanel project. The next time you set a user to "Signed Up", they'll get a GCM notification.