I’ve found the solution – I needed to unhide the “Excerpt” box in the Screen Options dropdown. The text I put in there appears on the home page, leaving the main page free for the actual content.
Sorry, I might have explained that badly. The fix above hides the text on the home page.
I would like to have the text which is show in the widget on the home page hidden on the page you go to when you click “Read More…”