I need to know exactly how the AJAX URL REQUEST works in wordpress

my code :

php (in function.php):

    add_action('wp_ajax_nopriv_wpa56343_search_widget', 'wpa56343_search_widget'); // for not logged in users
    add_action('wp_ajax_wpa56343_search_widget', 'wpa56343_search_widget');
    function wpa56343_search_widget()
    {
      global $wp_query;
      $search = $_POST['query'];
      
        $args = array(
            'post_type' => 'post',
            'post_status' => 'publish',
            //'posts_per_page' => -1,
            's' => $search,
            'posts_per_page' => 20,
            /*'tax_query' => array(
                array(
                 'taxonomy' => 'category',
                 'field'    => 'term_id',
                 'terms'    => '1',
                 ),
                ),*/
        );
        $wp_query = new WP_Query( $args );
     
      if($wp_query->have_posts())
 while($wp_query->have_posts()): $wp_query->the_post(); ?>
            
    <h1>do some html work</h1>          
          
        <?php endwhile;
            wp_reset_postdata();
              //endif;
              die();
    }

my simple SCRIPT code

$(document).ready(function(){  
      $('#s-mobileapp').keypress(function(event){
          
           var query = $(this).val(); 
                      
           if(query.length > 2)  
           {
                $.ajax({
                    type:"POST",
                    url: "./wp-admin/admin-ajax.php",
                    data: {
                        action:'wpa56343_search_widget_mobileapp', 
                        query:query,
                    },
                    success:function(data){  
                        $('#searchwidgetresult-mobileapp').html(data);
                        $('#searchwidgetresultcontainer-mobileapp').css({"display":"block","border":"1px solid rgb(187, 187, 187)","box-shadow":"0px 2px 9px rgba(0, 0, 0, 0.4)"});
                    }
                    });
           }else{
            $('#searchwidgetresult-mobileapp').html('');
            $('#searchwidgetresultcontainer-mobileapp').hide();

           }
      }).blur(function() {
                    setTimeout(function() {
                        $('#searchwidgetresultcontainer-mobileapp').hide()
                    }, 500);
                }).focus(function() {
                    $('#searchwidgetresultcontainer-mobileapp').show();
                });
});

The problem is these functions works good in some pages and some not.

when i test the SCRIPT code by alert i found that the code stops in the ajax url call

  url: "./wp-admin/admin-ajax.php",

so is there any other way for this url method ?

something like public ajax call ?