Hi Aleksandr,
Seems like there is a time limit session on your Cpanel since when I'm trying to navigate through your site files it will prompt me with this message again,
http://take.ms/bUGy2 . To be honest this will be very difficult for us to debug since we don't have the exact idea on why the error occur when trying to attach the attachments and we are not able to dump the error information if we don't have the access to the core file.
Is it possible for you to temporarily turn off this authentication or perhaps make the session longer so we don't have to keep bugging you with the authentication code each time the session is expired.
Looking forward to your response!