Create onClick Event to Re-load a Widget

I think you something like this.

HTML
Add this in widget

<a href="https://wordpress.stackexchange.com/questions/254474/javascript:void(0)" id="add_my_fav">Add To Favourite<span id="favCount"></span></a>

Ajax
Add this in template

<?php $nonce = wp_create_nonce('addToFav'); ?>
<script>
jQuery(document).read(function(){
    jQuery.ajax({
        url: 'http://example.com/wp-admin/admin-ajax.php',
        data: {action:"add_to_favourite", secret:"<?php echo $nonce; ?>"},
        method: 'POST',
        success: function(res){
            jQuery('#favCount').html(res);
        }
    })
})
</script>

PHP
Add this in functions.php

function addToFavourite(){
   if(isset($_POST['secret']) && wp_verify_nonce($_POST['secret']))
   {
       // Add here your code to add favourite

       // return new count value
   }
}
add_action('wp_ajax_add_to_favourite', 'addToFavourite' );
add_action('wp_ajax_nopriv_add_to_favourite', 'addToFavourite' );