Hi Maxim,
the error message suggests that you are trying to send the email as a different sender than the one which owns the mailbox (e.g. 'mail-service@MyCompany.com'). Could you please check on that?
If you are sure that you should really be able to send through the 'mail-service' account with a different sender, then it is a matter of configuring the SMTP server in a right way to allow this as it depends on the SMTP server whether it will accept message with a 'different' sender or not.