Automaticly add slug to posts

You can do this by setting 'with_front' => false while registering the new custom post type, To remove the custom post type slug which WordPress adds by default, pass 'slug'=>'' empty.

E.g.

//Example - how to pass array to rewrite
$args = array(
    // This rewrite settings will remove the slug you want also the slug - /custom_post_type/
    'rewrite' => array('slug'=>'','with_front'=>false),
); 
register_post_type('custom_post_type',$args);

Reference – Register_post_type()