How do I go about fixing this apparently messed up upgrade?

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The first thing to try:

  1. Manually download a fresh copy of WordPress
  2. Log into the FTP account and upload everything EXCEPT the wp-content folder

It is important that you do NOT copy over the wp-content folder or the wp-config.php files. These are custom to each site.

This will allow you to make sure the upgrade is complete.

If that doesn’t work, you should remove the .htaccess file from the site, and see if you can get access back to the admin.

Also… Godaddy is on my short list of hosts that I won’t work with on WordPress sites, but it does generally work with very basic site setups.