RESTful API, tracking client's individual user requests

I have an API set up on my server (node.js) which potential clients can send request to. One such client would like for me to set up a structure where they would pay only for the amount of their users who would connect to my API. They are creating a mobile application. Regardless mobile or web, I'm not sure how I would be able to track their individual users, to make sure it is their users who are sending requests to my server and not the client himself. The client can make one request and send to 1000 of their users instead of 1000 of the client's users directly connecting to my server.

The only feasible solution I can think of is creating a plugin which they would insert into their app and it would connect directly to our server, bypassing the client's server. Something like a Facebook Share/Like, Google +1 button. Creating the plugin would require to create the entire request and dynamic layout in each platform language, which is outside my scope at the moment.

Is there any way to have the end-users connect directly to my server through the client's app, bypassing the client in the middle, allowing me to know how many users will be connecting?

via denikov