Thanks.

I used the Maintenance Plans wizard to create the backup jobs and I see in there that it contains the following for each db:

DECLARE @preferredReplica int

SET @preferredReplica = (SELECT [master].sys.fn_hadr_backup_is_preferred_replica('TestDB'))

IF (@preferredReplica = 1)

BEGIN

I found that the backup job was failing and sending out errors on the secondary server if I had "Verify backup integrity" checked.