The article "a" places the focus on the specific payment being made, usually in a series of payments:
"I made a payment only yesterday!" "I make a payment only when I think of it".
Without the article, the focus is more diffuse, more general, and may even imply full payment:
"You should make payment immediately." "I made payment as soon as I received the invoice."
As is often the case, though, there is no hard-and-fast rule about which should be used when ...