The message does not appear completely but seems to be “550 5.7.1 Client does not have permissions to send as this sender”. Obviously the user does have the permissions (the user was the owner of the mailbox itself).
Run AdsiEdit and select the following path:
Configuration – Services – Microsoft Exchange – <MY_ORG_NAME> – Administrative Groups – Exchange Administrative Group – Servers – <MY_SERVER_NAME> – Protocols – SMTP Receive Connectors:
Properties of the “Client Proxy <MY_SERVER_NAME>” receive connector, “Security” tab and grant the “Accept Authoritative Domain Sender” allow permission for the “Authenticated users”.
Restart server