How to get certain part from widget

You cannot separate the markup from the output in this widget. The links are built from the result of mlp_show_linked_elements(), so you can use that to create the elements with a custom function or a new widget.

This will print the links to all connected pages:

mlp_show_linked_elements(
    array(
        'link_text'         => 'text',
        'show_current_blog' => TRUE
    )
);

You can add a parameter 'echo' => FALSE to use the result as a value for a variable:

$links = mlp_show_linked_elements(
    array(
        'link_text'         => 'text',
        'show_current_blog' => TRUE,
        'echo'              => FALSE
    )
);
// do something with the links …
echo $links;