The idiom "on someone's back" usually means that someone is constantly nagging or pressuring another person to do something. Some of the synonyms that could be used to convey this idea include "riding someone," "hounding someone," "pestering someone," "bothering someone," "nagging someone," "badgering someone," "plaguing someone," or "harassing someone." All of these words could paint an annoying and persistent picture of a person who won't give up until they get what they want. While this behavior might be necessary at times, it could also be harmful if it's done excessively or without consideration for the other person's feelings.
