Well.... This is embarrassing......

So, I DO only have the one database for the site. When I made the backup, I went through phpMyAdmin.
When I restored it, I tried to do this through Plesk, and it clearly didnt work even though Plesk said that it did.
So.... Now I tried importing through phpMyAdmin I get the following error:
#1046 - No database selected
When I try to import through Plesk, I get a different error:
ERROR 1062 (23000) at line 43: Duplicate entry '1' for key 'PRIMARY'
program 'mysql' finished with non-zero exit code: 1
I realise that its my account causing the problem in the second case.
Any suggestions to solve this?
Many thanks.
EDIT - I am going to try a fresh install, and use the same prefix as the backup, then remove the entire database and create a new one to house the backup......