Hi Dan, it appears to be an issue with KB2588516. What you can do is download the standalone installer for that update and try to install it manually. Since I am not sure you Windows version I am hesitant to link to one. A quick google of KB2588516 Windows Version download should take you to the correct .msu file you need. Once downloaded manually extract the contents.
You can find a tutorial
Windows Update - Extract an Update on how to extract and update an msu manually. Once you get that finished shut down, then re-run the Update Readiness Tool. Your Checksur.log should be empty. If so you can safely download the remaining updates through Windows Update.