When you share a post on Facebook, or when Jetpack Publicize publishes a post to your Facebook page, Facebook crawls the page and looks for Open Graph meta tags in the head to build a complete post preview (with an image, title, description, …).
Jetpack automatically creates these Open Graph Meta tags for you unless you already use another Open Graph plugin, in which case we let the other plugin handle things.
I had a look at your site, and it seems that you’re using 2 different plugins to output Open Graph Meta Tags on your site. I’d suggest deactivating one of these plugins, as Facebook usually doesn’t really like duplicate tags.
We can now start looking at what Facebook sees when browsing your posts. You can use Facebook Debugger for that:
https://developers.facebook.com/tools/debug/og/object?q=http%3A%2F%2Fwww.horsereporter.com%2F2015%2F06%2F10%2Fchepstow-hosts-diar-prep-race%2F
https://i.wpne.ws/bbz6
As you can see, Facebook seemed to have some issues downloading the images you had defined in your Open Graph Image Meta Tags:
og:image was not defined, could not be downloaded or was not big enough. Please define a chosen image using the og:image metatag, and use an image that’s at least 200x200px and is accessible from Facebook. Image ‘https://www.horsereporter.com/wp-content/uploads/2015/05/pxOneWorld.jpg’ will be used instead.
I can now click on “Fetch new scrape information” to force Facebook to crawl your post again, and look for an image again. here is the result:
https://i.wpne.ws/bc21
Now Facebook complains about the duplicate set of tags, so you’ll need to get that fixed. I’d suggest starting with that, and then clicking on the “Fetch new scrape information” button again to see what Facebook sees when browsing your posts now.
Let me know how it goes.