The story presents power as the ability to command compliance or resist compliance, and specifically, to determine whether or not the throat exam will take place. This power is important for both the doctor and for the child, Mathilda.
For the doctor, who typically relies on his own authority and his gentle coaxing manner to get things done, power comes easily. And when someone like Mathilda comes along who asserts her own power to resist him, shutting down his own power, the doctor flies into an unreasonable fury. For him, having the power to perform his work normally, for his patients to comply with his requests and instructions, is the basis of his efficacy as a doctor and may even form much of his self-concept as a respected adult and as a man. Threats to his power, such as those presented by Mathilda’s resistance, are so disturbing to the doctor that he stops acting reasonably and begins to rely on his own superior physical force in order to assert his power.
For Mathilda, the obviously beloved and beautiful child, power sometimes comes easily and sometimes does not. She can control her parents by lying to them, telling them her throat doesn’t hurt when it truly does—thereby keeping her potential disease a secret and (in her childish mind) keeping it from becoming real. She can also to some extent control her parents and the doctor by ignoring their commands and by physically fighting back against them, often in a necessarily sneaky way (which the narrator describes as “catlike”). But her power meets its limit when the adults exert physical force on her, something that she probably is well aware of—and that’s why it’s so important for her to have her own means of resisting compliance. Because she’s young, because she’s female, because she’s dealing with the uncertainty of possibly having a fatal disease, and because she’s constantly under the control of others—parents, teachers, even the doctor, who’s a stranger to her—she must find a way to maintain some sense of power, or else she will feel lost and frightened.