Are you logged in as the MW admin or some extra user that you have set up? If the latter, it's most likely the group permission setup that is the problem.
(our Mailwizz solution is very different from the rest of yours so I can hardly tell how a standard installation looks these days)
Thanks guys. After enabling 'exec function' in the PHP of my new server, a delete button magically appeared. Now, after deleting the customer I wanted to remove, I find a new message that says "Pending delete". How do I finish this off?