Call a PHP function with Ajax when the user clicks a button in wordpress

Pass values to your ajax request like this

jQuery(document).ready(function() { 
   jQuery(".mark-as-read").click(function () {
    console.log('The function is hooked up');
    jQuery.ajax({
        type: "POST",
        url: "/wp-admin/admin-ajax.php",
        data: {
            action: 'mark_message_as_read',
            // add your parameters here
            message_id: $('.your-selector').val()
        },
        success: function (output) {
           console.log(output);
        }
        });
    });
});

The PHP code to handle the request (here too you should handle the nonce in real life code):

// register the ajax action for authenticated users
add_action('wp_ajax_mark_message_as_read', 'mark_message_as_read');

// register the ajax action for unauthenticated users
add_action('wp_ajax_nopriv_mark_message_as_read', 'mark_message_as_read');

// handle the ajax request
function mark_message_as_read() {
    $message_id = $_REQUEST['message_id'];

    // add your logic here...

    // in the end, returns success json data
    wp_send_json_success([/* some data here */]);

    // or, on error, return error json data
    wp_send_json_error([/* some data here */]);
}

Credits: https://wordpress.stackexchange.com/users/52894/mike