I am now also facing this issue for the second time. To me it appears to be linked to chaning the primary domain. That seems to cause the Rest API to break.
Last time my workaround was to make a backup in wordpress, remove the site from Enhance, create a new site in Enhance, restore the backup in wordpress.
Not sexy, so a fix to the root cause would be appreciated 🙂
Ps. I'm not running cPFence, it is a pure Enhance server.
Edit: SOLVED with Adam's help. It seems changing the domain somehow removed a rewrite in "Developer Tools" > "Nginx" > "Rewrite". Adding back / to index.php resolved the issue.