Hi,
You have a cache issue on your website.
Follow that procedure, and after you've confirmed that the cart is still empty on the homepage, just add a parameter like ?test=1 in the URL of the page in the address bar of your browser and validate and you'll see that the cart module now has the correct content.
And if you remove again the parameter and display the homepage again, you'll see again the cart module without the added product.
So what that means is that the cart in the database has the correct data, but that when you go to a page you already displayed, the website / server doesn't ask HikaShop to display the cart module because that URL was already generated previously and thus you don't see the new content of the cart.
So the question now becomes : what cache mechanism do you have in place on your website / hosting which could do that ?
You'll likely want to tune that cache mechanism to no cache the cart module or pages where the cart module is displayed if the cache system has such settings, or disable that cache mechanism altogether and replace it with one with more fine-grained control of what to cache.