When a High Availability (HA) environment failed over, on the node that became Secondary, the backup jobs did not gracefully handle the change. The Maintenance Plans are supposed to intelligently determine whether each database is Primary or Secondary in an HA environment, and skip the backup if it is not Primary. This check actually fails (practically speaking) if you have "Verify backup integrity" checked.
When using AlwaysOn, you must connect using your listener name, not your node name. If you connect with the node name, it will only work until there is a failover, which would defeat the purpose of your High Availability setup, right?