Is there anyway to remove these links from the output of the media module? I suspect that they're responsible for breaking the link tag I have wrapped around the Media image in the WYSIWYG editor.
On the rendered template of the page this looks like
How can I prevent these links from being embedded by the media module? They even appear to show when I am not logged into the Drupal administration too.
You'll have to change the sort order of filtering within your text format, admin/config/content/formats/full_html (for Full HTML text format). Just place the "Convert media tags into markup" before the "CKeditor Link filter".
There is a patch here https://www.drupal.org/node/2401811
This behavior is a regression introduced by another patch that may well get rolled back.
I posted this as an issue on the media_wysiwyg module bug tracker over at https://www.drupal.org/node/2357993 and have been working my way through it since. Here is the best result I have managed to achieve so far copied from my responses to that issue.
This issue is mostly caused by the file_entity module. To work round this I have restricted the view modes available to media_wysiwyg in /admin/config/media/wysiwyg-view-mode. I have set images to only allow the WYSIWYG view mode like so:
I have then created an override template for the file entity called file--image--wysiwyg.tpl.php (see https://www.drupal.org/node/1793548) and I have the following code in it:
<?php
hide($content['links']);
print render($content);
This has now removed all the annoying links, headers and wrapping <div>s from the output EXCEPT for the media_wysiwyg wrapping mentioned in comment #4.
As this is just one wrapping <div> and the page is HTML this now no longer breaks the link, but it would still be preferable to be able to remove this annoying wrapping <div>.