YES! You can change each page to have a custom layout for your mobile site. Or they can have the automatic layout. Here is how to make changes:
FIRST
Go to the page that you want to appear different on your mobile site. Click on the "Mobile Version".
See screenshot:

Then you can choose to edit this page manually.
see screenshot:

SECOND
Click on Save and start editing your page for your mobile site.
This will start a blank page for your mobile version. You can copy and paste content from your regular page to your mobile page. This will not interfere with the current page that you have laid out for your desktop/regular page.