How can I set up payment confirmation notifications for customers?

Hello everyone. First of all, sorry for my bad English.
My store has two manual payment methods. There is an integration in Shopify, allowing the order to be marked as “paid” when the costumer actually pays it, outside of Shopify.
The problem is that the costumer doesn’t get any notification after he pays the order, because the default Shopify notification (order confirmation) is actually sent before the payment is done.
can anyone help me on getting a script to make it possible for the costumer to get this missing notification to make him know the payment is ok and we’ll work on the order preparation?
I hope my English made this clear. Thank you so much! Helder, from Portugal.

Hello. Which file? There is no file.