Term for "to have a hidden purpose for being involved in something"?

Consider using hidden agenda:

An undisclosed plan, especially one with an ulterior motive.

[American Heritage Dictionary of the English Language, Fifth Edition]

As @Phil Sweet commented, you could also consider using "personal agenda" or just "agenda". I've just found a related question, Meaning of “have an agenda”.

Consider ulterior motive:

If you say that someone has an ulterior motive for doing something, you believe that they have a hidden reason for doing it ⇒ Sheila had an ulterior motive for trying to help Stan.

Secret intention or hidden ploy.

This is what happens when a hidden ploy is unveiled; it can never work the same way again, and can never have the same meaning or payoff for the perpetrator (The New Way to Compete: How to Be a Winner in Your Career and in Your Life, Harry A. Olson, excerpt on G. Books)