Cannot add edit themes and add plugins after multisite update

This is a wildshot but generally this happens when the correct version of jQuery is not loading for the dashboard.

While logged in you could check this in your browser. I.e. in Chrome right click and choose inspect element – go to the console tab and see if there are any undefined errors thrown this will confirm the issue.

Or, in the Elements tab search for jQuery and ensure the correct version is loaded.

WP 3.9 should have jQuery V1.11.0
WP 3.8 should have jQuery V1.10.2

We can further help confirm JS is the error by placing: define('SCRIPT_DEBUG', true); in your wp-config file in the root of your install and see if your issue “goes away”.