THE SOLUTION.
The problem with sending e-mail at GoDaddy hosted sites is . . . .
You can't send an e-mail through their SMTP servers that is FROM a Domain outside their DNS Servers.
For example, if your domain is
www.fred.com and that domain is controlled by GoDaddy servers then you CAN send an e-mail with this FROM address, for example
fred@fred.com.
If, however, your FROM address is
fred@gmail.com or any other domain outside GoDaddy's DNS control then the mail will be denied.
Makes sense if you think about it . . . it's how GoDaddy prevents people from using their servers to spoof e-mail.
How do you avoid this - by making sure that every admin/super user account and/or the HOST e-mail account ( in HostSettings) are set to GoDaddy e-mail addresses.
Since GoDaddy gives you 100 e-mail accounts with almost all of their hosting accounts, just set one or more of them to something like
webmaster@yourdomain.com, then set them to relay to the address you really want.
That and using the relay-hosting.secureserver.net with annonomous authentication and you're all set.