Generally, email servers look to verify that the sender is a customer of the email provider. Possibly, if you post your server settings and your SMTP server settings, some suggestions will surface. Thank you.

If you're sending through the gmail smtp from a non-gmail account, gmail will automatically switch the sending account with the gmail address, so recipients see the message as being sent from the gmail address. Open gmail settings on gmail.com, and in the Accounts & Import section, Send mail as, add the non-gmail account.

Thanks for pointing me to the Accounts & Import section, but I'm still tearing my hair over this. I'm actually looking to do pretty much the opposite of what you said.

You said "If you're sending through the gmail smtp from a non-gmail account, gmail will automatically switch the sending account with the gmail address, so recipients see the message as being sent from the gmail address."

The main thing is: I want recipients to see the message as coming from my domain, NOT from my Gmail account.

Please reread sfhowe's post. He explained how to make the desired correction that you seek.

I'm stuck at this screen. There are no SMTP servers for my domain name, so I can't fill it in. When I was using Plusnet for my email, it was set up in such a way that outgoing emails showed my domain name address as sender, but I haven't been able to do that with Gmail.

Is there a workaround?

Instead of adding your domain account as a sender in gmail, you should add an smtp server to TB with your domain settings, e.g. it might be something like: smtp.yourdomain.com, 465, SSL/TLS, normal password, email address. Then, select the domain account in the left pane of Account Settings and point Outgoing Server (SMTP) in the right pane to the domain smtp.