How can I pass value to function in add_menu_page?

Don’t use a function for the callback, use a class. Something like this:

class Competition_Manager_Page
{
    private $extra;

    public function set_extra( $value )
    {
        $this->extra = $value;
    }

    public function render()
    {
        // show you page content, then:
        print $this->extra;
    }
}

You can extend the class’ functionality later.

Now, when you register the menu, create an instance of that class first, and add you extra data:

$page = new Competition_Manager_Page;
$page->set_extra( 'Test' );

add_menu_page(
    'Competition Manager 2013-2014',
    '2013-2014',
    'manage_options',
    'manager_2013_2014',
    array( $page, 'render' ), // replacement for the former function
    plugins_url( 'competition-manager/images/cup.png' ), 81
);