M+Google team continues to receive reports that Mail.app on Mavericks (10.9.x) is still having issues with Google Mail. The ITS Service Desk has received a few tickets in which Mail.app is not updating after first opening (confirmed by Apple) or, in some cases, it is deleting email. It is highly unlikely deleted mail can be restored, even with a ticket to Google. At this point, we recommend not using Mail.app on Mavericks to access your M+Google account due to these issues.
If you have a support issue relating to this, please contact the ITS Service Center.