In previous versions of Clean Text there was a "Iconized" window functionality with the option to clean directly the Clipboard content.
In current version that feature is present in an icon in the menu bar. If you can't see it, please update your version of the app.
In addition to this, you can also find actions to process Clipboard text in the sidebar of the main window of the app. Just use those items and all the text present in the general clipboard of the Mac will be processed. In this way you don’t need to paste text and then to copy it from the main window.