When I think about this question, it simply goes back to one major design choice that is taking place right now and that is minimalism. More white space, collapsable menus, and simple text features.
So I would suggest that the homepage simply feature that the main story with it’s attention grabbing image and a snippet from the article that users can click on if they want to read more. Then I would suggest to get rid of both the left and right columns which are both animated or has moving picture. These things pull my attention away from that main story in the center of the page. I think it’s dizzying and unnecessary information that can either be pushed further down on the page as part of one column or be on a separate page that a user can click to.
Another suggestion would be to keep the main column one size in width. Some of the information (such as the recommendations section) in that column is larger in width and kind of breaks up the page instead of flowing with the rest of the text and being flush to one size of the column. This is distracting as well as my eyes are drawn to the information that is stick out amongst this main column instead of reading the information in order of the visual hierarchy.
My last suggestion is one that find common amongst sites and that is the placement of the sharing logos. These tabs are important to almost all sites these days as sharing through social media or email allows for more traffic on the site. This is even more important to publications as it allows for the stories to get shared and read by more than the people who visit the site. On this site, they are placed only at the top or bottom of the article. That forces the user to scroll to either end of the reading to share the article which is not mobile conducive when you have a long article. So my suggestion would be to place those sharing buttons along the side of the browser and have it scroll continuously like a lot of site’s I’ve seen to allow the user to more easily share the articles they read. That can be a big help to the readers who want to share.