Take a look in index.template.php in your theme directory.
Down near the bottom, you'll have the function that spits out the menu tabs, which is where you put your new entry.
What'cha need to do is throw an IF around it, with the permission you created. There's two ways to do this, which you'll see as you inspect other php files -
if (allowedTo('myPermission'))
{
spit out my menu item;
}
-- or --
if ($context['myPermission'])
{
spit out my menu item;
}
Take a look at how the other menu entries did it, a good example would be the Calendar item. Obviously, you'll need to repeat this with any other themes you have installed.
Also make certain to assert that the user has the permission, in your mod your mod itself, in case they call it directly. Somewhere up top, you'll probably wish to throw in a
isAllowedTo('myPermission');
...before they are able to do anything bad.
Note that allowedTo( ) returns true or false; isAllowedTo( ) will invoke the "F** Off!" screen should the user not have the right.