Hi,
Thank you for clarifying. It sounds like you are adding an order manually. The emai with the link that says Pay for this order
links them back to the site successfully.
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Link to image: https://cld.wthms.co/fGXt0u
You are saying that when the customer pays for the item with PayPal that the transaction does not go through and they are brought back to the site.
Just for reference, the cart is not a part of this process, in fact there could be items in the cart that have nothing to do with the order being paid for. What is being paid for is an order.
The URL that you are taken to from the email should look like this.
https://yourwebsite.blog/checkout/order-pay/498309/?pay_for_order=true&key=wc_order_VYyyzYT72Hpce
The page should say Pay for order like this.

Link to image: https://cld.wthms.co/4FgZn0
Can you confirm that this is true on your site?
If so, then the issue sounds like it has to do with PayPal and not WooCommerce. If you are taken to the PayPal site and then sent back to your store without payment being taken, the reason for that should be on the PayPal side. Can you check for any orders, failed or otherwise, on the PayPal side.
Can you also check the orders in your store and see if the order notes were updated at all? Anything about failed payments, or IPN not working, or are the order notes not reflecting that anything happened?
Without hearing back from you, my educated guess is that something on the PayPal side is not working, which they would be able to help you with. The other thing that could be happening is that either the theme or a plugin is conflicting and what is being sent to PayPal is not quite right. But this is just guessing before seeing your answers to the above questions and troubleshooting.
Cheers!