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Relevant part of the log file:

DEBUG   |20230416 02:27:16|Akeeba\Engine\Core\Domain\Db :: Iterating next database
DEBUG   |20230416 02:27:16|Akeeba\Engine\Dump\Native :: New instance
DEBUG   |20230416 02:27:16|Akeeba\Engine\Core\Domain\Db :: Now backing up 21EDCE09-F377-4D59-92A3-CB5BD6160F1E (bganzac_ForumgiJQv5pjuAG9YyvM8)
DEBUG   |20230416 02:27:16|Akeeba\Engine\Dump\Native :: Processing parameters
DEBUG   |20230416 02:27:16|Kettenrad :: Domain object returned with errors; propagating
ERROR   |20230416 02:27:16|Akeeba\Engine\Dump\Native :: Database Error: Could not connect to MySQL

Diagnosis:

You have accidentally set up an additional database with no connection information or invalid connection information.

Since the connection information is invalid / missing, Akeeba Backup cannot connect to it, therefore it cannot back it up.

Failure to back up a database constitutes a failure to complete the backup, hence your receiving an error about the backup not running.

As a result, we need to remove the erroneous database entry.

Solution:

Go to your site's backend.

Click on Components, then Akeeba Backup.

Make sure backup profile #1 is selected.

Scroll down to “Include and Exclude Information”.

Click on Multiple Databases Definitions

Click the trash can icon on all rows which do not correspond to databases you want to back up.

Do remember that Akeeba Backup always backs up the main site database, i.e. the database the connection information for which you have set up in Joomla itself.

You sent me your reply by email:

Thank you for your reply 

No we didn't accidentally set up a secondary database this site has a bb3 data set associated with the main site

This was working fine until some error happened about two years ago and the client has only just made us aware of it 

I believe the database is set up correctly however the site is now woefully out of date and as we don’t have any backups I am unwilling to just plough on and corrupt the site further

You can always create a copy of the backup profile in Akeeba Backup, Profiles.

Then, you can activate the new profile (in the main Components, Akeeba Backup page) and go to Multiple Databases Definitions.

You can now edit this database definition to reflect the real connection information for the phpBB3 database. You will find this information in the config.php file of the phpBB3 installation, see https://www.phpbb.com/support/docs/en/3.2/kb/article/rebuilding-your-configphp-file/ to make sense of what you are seeing.

If the new backup profile doesn't work you can always delete it and start over.