I have a problem with my Zumo 350 too!
Just yesterday installed my Zumo 350 with hard wire kit to my battery on my BMW F650 twin. Drove 150 miles today and twice I got a message that it had lost external power. I squeezed the unit into the cradle and it seemed to work after each time I did this. I don't consider this a solution but it must be as others have said, something to do with the pin contact. I emailed Garmin today and will wait to see what they say about it.