How would you test a function that calls a third party payment API?

Isolate the boundary. Put the HTTP call behind a thin client, then in tests replace it with a fake or patch it where it is used, not where it is defined. Use pytest fixtures for setup and parametrize for the success, failure, and timeout cases. Add a small set of contract tests against the real sandbox so mocks cannot drift.

Testability is a design question first. If the payment call is buried inside business logic I pull it out into a client class with a couple of methods, and then the logic takes that client as an argument. At that point most tests need no mocking library at all, because I pass a small fake that returns canned responses, which is easier to read than a stack of patch decorators. Where I do patch, I patch the name in the module under test, not in the library module, because patching where it is defined does nothing once it has been imported elsewhere. Fixtures build the objects, and parametrize covers the interesting cases: approved, declined, timeout, and a malformed payload, since the failure branches are what actually break in production. I also keep a handful of tests against the provider sandbox, tagged so they run nightly rather than on every commit. That caught a change where the provider started returning an error code we had never mocked.