You can do this in business.google.com, click on the posts tab, then at the bottom right of the post is a 3 dot, click that and you can edit, delete or see on search.
@djbaxter, if you go to the GMB dashboard -> "Posts" and look in the lower-right corner of a specific post, you'll see an ellipsis. Click on the ellipsis and you'll see the option to edit or to delete.