I'm trying to send an email with a source address in my hosted email domain to a destination address also in my hosted domain, and I'm relaying though a third party (or though you, without authenticating). Why does the mail not arrive?
If you wish to send email to an address that we host, from an address that we host you will need to authenticate against our server before it will be accepted. Since our server knows the source is a local address it wants to be sure it is a valid message. It's a protection we use to try to prevent Email spoofing (in this case where someone remote appears to be from a local domain), which is often done to trick someone. You can imagine the potential problem when a message from 'firstname.lastname@example.org' goes to 'email@example.com' but is actually originated outside the organisation and is unauthenticated.
What you are likely doing, is sending your email from and for/to and address in your domain hosted with us using your other mail provider, which accepts it, but it then gets rejected when our mail server receives it from them.
An analogy would be asking your friend to deliver a letter with his own name on it as the author to your other friend.
There are 2 basic solutions.
1)You can add your mail account (along with authentication) to your mail client software so that it sends though us when sending email from your domain.
See our details on outbound mail in the basic client email settings. https://support.enlightenhosting.com/hc/en-us/articles/205553113-Basic-e-mail-client-settings
2) Use a different source address, one which our mail servers are not authoritative over. The email will then be treated as any other external, inbound email.
Note a variant of this is if you're trying to send from yourself (or in fact anyone) to any third party though us without authenticating. In that case you'll get the 550 "we do not relay" response. This is simply to prevent our severs being used as a public relay and affecting your Internet reputation.