The same consolidated update history report that you linked an image to above has a column for ‘Result’, which will show failures. That said, I’m unsure how useful this would end up being.
If you use BatchPatch to perform an update installation, BatchPatch will report in the ‘Remote Agent Log’ for that task the details of each update installation (success, failure, etc). The ‘Remote Agent Log’ column in BatchPatch is also saved automatically to the remote agent log file that is stored on the target computer (default location on target is C:\Program Files\BatchPatch\batchpatch.log). You can use the HTML export feature, mentioned in my previous posting, after an update installation task to review the failures. And of course you can review the batchpatch.log file on a target computer to examine the failures from previous tasks. However, aside from the update history report mentioned at the top of this posting there is not another way to pull a “live report” of updates installation failures that took place in the past.