For the Paypal express / Amazon pay checkout issue, our app creates a draft order with the appropriate discounts, and so shoppers must complete checkout for the draft order our app creates in order to get their discounts.
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When the Paypal / Amazon button is clicked on the cart page, shoppers are instead taken to a separate checkout process for a non-discounted order.
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Unfortunately, we don't yet have a perfect solution to this. What we recommend is hiding the Paypal / Amazon buttons on your cart page but keeping them in your checkout. So long as shoppers click the Paypal / amazon button in checkout, they'll receive their discounts.