Hi James,
I don't really know what Google is doing here. If a page no longer exists then it should remove this from its index after a while. To make the process smoother you could consider publishing a redirect page at this location which sends visitors to some other useful information? Otherwise just wait until Google reindexes.
Regarding the broken link. That link does actually exist on that page, but the link text is just a space so you can't see it in the wysiwyg editor! It's just under the Other Events title.
To remove it you could use the View Source option, scroll to the bottom and remove it. Or if you turn on Show Invisible Characters you'll see the blank lines under Other Events that you can then delete. When you publish the change the link will disappear from your Broken Links report.
Thanks
Joe