YES! you can make a site menu complete separate from your regular site menu. And they will not interfere with each other. Here's how:
FIRST
Enable the ability for a separate mobile site here:
YourDomain.com > Mobile Version Settings
See screenshot:

SECOND
Go to your Site menu area: YourDomain.com > Site Menu
See Screenshot:

Now you will have a drop down here that will allow you to select which version of your site menu you are working on.
By default it will look exactly like your Desktop mobile version. You can add and remove the site menu items without it actually removing any pages. This is just adding a link in your site menu to a page.