Friend has contacted me because he has sent an Email and received a " message undeliverable because the destination computer was not reachable within the allowed queue period". Ant help appreciated.

If this is an email received from the email provider, it means that eM Client has successfully delivered the message to the sender’s server, but the sever is replying that it was not able to deliver the email to the recipient’s server.

Probably haven’t explained properly: my mate tried to send me an Email several weeks ago but got the failure message above. He tried again today and even though he sent the message successfully to others he again got the message above from my address. His concern was that I may be missing other messages but I have not had anyone else complaining they could not get their Emails to me. 

If the message is from the email provider, that means that eM Client has successfully delivered the message to their server. That is all an email client can do. If that server cannot deliver it to another server, that is between his provider and your provider. You might want to ask them about this.

If the message is from eM Client in the form of an error, can you post the full error message?

Have asked my mate to send the message to me. I think he can use his wife’s account to do so but not 100% sure. will keep you informed. Thanks for helping. Murray.

Sent message to mate and, of course, it all worked and we can now Email each other at will. Thanks for your assistance and interest in my problem. Murray.