When I open a Private window my Evernote Web Clipper add-on disappears

Sorry, don't use it. My guess is that is it just doesn't work in Private Browsing.

You might want to try Evernote Community Support for an answer from other users. https://discussion.evernote.com/

The comments aren't rally hopeful

• https://addons.mozilla.org/en-US/firefox/addon/evernote-web-clipper/

You can check the Browser Console for error messages from this extension.

• https://developer.mozilla.org/Tools/Browser_Console