This error could be caused if you're trying to send through a network that is not your normal ISP, e.g. public wifi, but your outgoing server settings don't authenticate your credentials, e.g. the connection security is None or it sends on port 25.

What are your settings in Tools/Account Settings, Outgoing Server (SMTP) at the bottom of the left pane?

If the connection security is not SSL/TLS or STARTTLS, the solution would be to add an account with secure connections and then send through that account's smtp. Or, use webmail when relaying is not allowed.

I have tried the setting as ssl/tls and starttls and none and none make any difference. I have changed nothing since I downloaded the latest update, but I find I also can't send any emails.

If the problem started after the last TB update, it's probably due to an antivirus program that's incompatible with the new TB version.

What is your AV program, and do you have it scanning email, and who is the mail provider?