In S03E17 -
/ is the name of episode) of Person of Interest, Root undergoes a surgery at an otolaryngologist. Did the device operated by the Decima guy impact so badly that it broke her ear (hearing ability)? If so what did she give Finch to repair?
Root tells Finch:
Root leaves the earbud, which she has been using until Decima's countermeasure rendered it useless, behind.
I haven't seen the episode, but according to the Wiki:
I think this explains what her initial ear problems were. In this episode, she has a device surgically fitted to her ear to allow her to remain in constant communication with The Machine. As this is a fictional surgery, I'm not aware of its name!
Finally, from a different part of the Wiki:
So it appears she gave Finch a chip to help prevent the activation of Samaritan, then took it back from him.