Welcome to the Community and thanks for your post!
Sorry for the problems you're having sending emails when you're away from home. When you say you're having issues sending emails, do you mean in bulk or just to individual mailboxes?
Have a look to make sure all your settings are correct here >> What are the settings for outgoing and incoming BT Email servers? <<
Others should be along shortly.
In addition to RobbieMac’s comments, make sure that you have set the outgoing mail password, (which is the same as the incoming password).
Outgoing mail settings require password authentication. You can reach the outgoing mail settings page by selection of the outgoing account name.
This setting is easily missed.
Can you send and receive when at home and is this when you are connected to your BT Homehub?
If you can it would appear that it is not a password problem. If it was it would not work at home.
When you are out and about how are you connecting to the Internet? Is it via wifi or 3g?