The correct answer to this open question is the following.
You did not mention the book or the text you are referring to. Without information is difficult to know what you are talking about.
However, trying to help you we did some deep research and can comment on the following general terms.
First, you are referring to the book called "The Immortal Life of Henrietta Lacks Teachers' Guide," written by Rebecca Skloot, a science writer that has published many interesting articles on this subject in prestigious slots such as the New York Times magazine.
The challenges that Skloot faced in communicating with Deborah was Deborah's personality and resistance to talk with Kloot. Deborah was a difficult research subject herself because she did not trust Kloot and her research. Deborah is the daughter of Henrietta, and for 20 years never knew about the existence of her mother's cells and that medical and research laboratories were investing millions of dollars in developing medicines. That is why Deborah showed resistance to talk to Skloot. She did not have a clear idea about what has happened with the cells of her mother and the purpose of the research.
Hello there!
The question states which quote represents her independence let's analyze each quote.
A)“Habitually obedient to John, I came up to his chair: he spent some three minutes in thrusting out his tongue at me as far as he could without damaging the roots…” This quote is incorrect because it says she is obedient to John.
B)“You ought to be aware, Miss, that you are under obligations to Mrs. Reed: she keeps you: if she were to turn you off, you would have to go to the poorhouse.” This quote is incorrect because it says that she is under obligations of Mrs. Reed.
C) “What do I want? A new place, in a new house, amongst new faces, under new circumstances…” This is correct because she is talking about what she wants.
D)I had had no communication by letter or message with the outer world: school-rules, school-duties, school-habits and notions, and voices, and faces, and phrases, and costumes, and preferences, and antipathies—such was what I knew of existence. This is incorrect because it says she has no communication with the outside world therefore not independent.
In conclusion the answer is C! Good luck!