This particular message indicates that the asset being published contains a link to another asset that the system is unable to locate. Refer to the following output from the Broken Links section of a sample Publish Notification:
[Destination: Production Web Site] about/staff/directory
Reason: Asset does not exist.
In this case, the about/staff/directory Page being published contains a link to /about/staff/john.html:
<a href="/about/staff/john.html">John Smith</a>
which does not exist in the system.
The solution here is to verify that the link is pointing to an asset that is managed within the system. It is important to note that file extensions for Page assets are not added until the final steps of the publishing process. This means that while the Page asset /about/staff/john may exist, you should not include the file extension when referencing it since that is added automatically during the publishing process. The actual link should be rewritten as:
<a href="/about/staff/john">John Smith</a>
(notice that the file extension has been removed from the link's href attribute)