It depends on a lot of things, but let's say (for concreteness) you are using Java or something similar. I'd write classes to represent the request and response models, call them A and B. Then I'd write a interface/class with a method from A to Future<B>. That method would be mocked everywhere, and most of your code doesn't deal with the API nonsense. (B is the data you want not the link to get the data.)

I may or may not actually test the implementation of that method.

In short: wrap up your external dependency.