Removing a Blog from the Global Posts Feed

On nearly all of our sites “Recent Global Posts” widget displays all of the latest posts from all blogs. It’s a really powerful feature that helps build community. However, one slight issue is that it displays post titles and links to posts even if they are on private blogs (the posts themselves remain private as clicking on the link demands a login). A private blog might be a staffroom blog, or a child’s early years learning journey, etc. It would be much better if those posts and links