Images not showing correctly
Without a leading '/' (or '/foo/' if the site is in a subfolder) the browser will always to try to find the image relative to the current path, not to the root of the site. That's why your current code works on the home page, but not on other pages.

To fix it you just need to add the base path as well:

&lt;a class="mobile-brand" href="#"&gt;
  &lt;img src="&lt;?php print $base_path . drupal_get_path("theme", "my_custom_theme"); ?&gt;/img/logo-mobile.png"&gt;
&lt;/a&gt;