I am not sure who I am replying to or what I did, but it now works (after a fashion).
Thanks to BDAqua and Barney-15E for their help.
I went to webmail and deleted practically all the mail.
On the 'damaged' computer I deleted 1,000's of e-mails and it now works. I still have over 1,500 each in the 2 mail boxes.
(It has not been practical to identify the problem e-mail - so the rationale has been to just to delete 100's.)
We have more that 5 Apple devices on the same Mail accounts and only 1 computer has the problem.
To say that the Mail package is inconsistent is an understatement.
If I select (say) some e-mails from an In box and delete them, I then find them again in the same In box (after switching to another Mail box and back again).
To guard my sanity I have checked and checked this, I have restarted several times (Mail & computer). (Both Safe & Normal mode.)
Thanks again,
Nic