Error sending emails on Apple Mail

Did you already have a look at the general troubleshooting mentioned in this related article?

If you can’t send or receive email on Mac - Apple Support

I had to set up an app specific password for accessing the mail server for my 3rd party mail account in the email settings for my mail account provider.

I doubt that this is the problem but it's the only thing I can think of.

On the Mac Mail go to Mail/Settings/Accounts (Not Mail/Accounts) in the menu bar.

Highlight the problem account in the LH pane

In the RH pane in the "Outgoing Mail Server (SMTP)" bit click on the arrow at the end of the "Account" box, scroll to the bottom and click "Edit SMTP server List"

Make sure you've got the same number of SMTP servers as you have accounts - i.e. one per account - and that the settings for the problem account are correct.

Apologies if you've already done this.

I'd start with DNS. Check the DNS servers on your Mac and your iPad and see if they are the same.

Is the iPad on the same network as your Mac or is it using mobile data?

Having said that, I still think that there's a chance that on the Mac the SMTP account isn't properly associated. When you get the "Cannot Send Message" dialogue box what happens if you select one of the other SMTP accounts and send using that?

Are you sure that the SMTP server is associated with the account you are sending from? That would also explain the work address thing - the account is associated with the wrong SMTP server. Try deleting the account and setting it up again.