Today, 21 Nov 2021. Magento 2.4.3

The issue is: When a customer applies a coupon code to add a new item on a checkout page, everything works perfectly, regardless of the count number of cart items on a checkout sidebar that is not updated. My helpful screenshot

Problem-solving:

A problem is the count value is not calculated on Observable Object ( KnockoutJS object).

Check files:

JS Component vendor/magento/module-checkout/view/frontend/web/js/view/summary/cart-items.js

Template vendor/magento/module-checkout/view/frontend/web/template/summary/cart-items.html

The count number is got from a function getCartSummaryItemsCount of a view component Magento_Checkout/view/summary/cart-items.

/**
         * Returns cart items qty
         *
         * @returns {Number}
         */
        getItemsQty: function () {
            return parseFloat(this.totals['items_qty']);
        },

        /**
         * Returns count of cart line items
         *
         * @returns {Number}
         */
        getCartLineItemsCount: function () {
            return parseInt(totals.getItems()().length, 10);
        },
But you can see a property totals is not an observable object.

 return Component.extend({
        defaults: {
            template: 'Magento_Checkout/summary/cart-items'
        },
        totals: totals.totals(),
A solution is: Find a text /*FIXED HERE*/ to know what I did fix it!
/**
 * Copyright © Magento, Inc. All rights reserved.
 * See COPYING.txt for license details.
 */

define([
    'ko',
    'Magento_Checkout/js/model/totals',
    'uiComponent',
    'Magento_Checkout/js/model/step-navigator',
    'Magento_Checkout/js/model/quote'
], function (ko, totals, Component, stepNavigator, quote) {
    'use strict';

    var useQty = window.checkoutConfig.useQty;

    return Component.extend({
        defaults: {
            template: 'Magento_Checkout/summary/cart-items'
        },
        /*-------- FIXED HERE ---------- */
        /* previous : totals: totals.totals() => remove "()" to keep a observable object */  
        totals: totals.totals,
        /*---------END- FIXED HERE -------*/
        items: ko.observable([]),
        maxCartItemsToDisplay: window.checkoutConfig.maxCartItemsToDisplay,
        cartUrl: window.checkoutConfig.cartUrl,

        /**
         * @deprecated Please use observable property (this.items())
         */
        getItems: totals.getItems(),

        /**
         * Returns cart items qty
         *
         * @returns {Number}
         */
        getItemsQty: function () {
            /*------ FIXED HERE -------*/
            /* previous : return parseFloat(this.totals['items_qty']); */ 
            return parseFloat(this.totals()['items_qty']);
            /*------ END - FIXED HERE -------*/
        },

        /**
         * Returns count of cart line items
         *
         * @returns {Number}
         */
        getCartLineItemsCount: function () {
            return parseInt(totals.getItems()().length, 10);
        },

        /**
         * Returns shopping cart items summary (includes config settings)
         *
         * @returns {Number}
         */
        getCartSummaryItemsCount: function () {
            
            return useQty ? this.getItemsQty() : this.getCartLineItemsCount();
        },

        /**
         * @inheritdoc
         */
        initialize: function () {
            this._super();
            // Set initial items to observable field
            this.setItems(totals.getItems()());
            // Subscribe for items data changes and refresh items in view
            totals.getItems().subscribe(function (items) {
                this.setItems(items);
            }.bind(this));
        },

        /**
         * Set items to observable field
         *
         * @param {Object} items
         */
        setItems: function (items) {
            if (items && items.length > 0) {
                items = items.slice(parseInt(-this.maxCartItemsToDisplay, 10));
            }
            this.items(items);
        },

        /**
         * Returns bool value for items block state (expanded or not)
         *
         * @returns {*|Boolean}
         */
        isItemsBlockExpanded: function () {
            return quote.isVirtual() || stepNavigator.isProcessed('shipping');
        }
    });
});