chip on one's shoulder

to have / wear/ to carry / a chip on one's shoulder (about sth) and to go about with a chip on one's shoulder (to bear a grudge; to be ready to be angered; to have a bad attitude or an attitude problem; to be sensitive about something that happened in the past and become easily offended if it is mentioned because you think that you were treated unfairly) — нарываться на драку / конфликт; искать повода для ссоры или драки, держаться вызывающе (затаив обиду, зло); быть задирой; ~ лезть на рожон; ~ лезть в бутылку

Example 1: John often gets into fights because he goes around with/carries a chip on his shoulder. (САм)

Example 2: He lost his game this morning, and now he has a chip on his shoulder.

Example 3: Watch out for that guy, he's got a chip on his shoulder.

Example 4: He has a real chip on his shoulder about being adopted. (OALD)

================
A person who has "a chip on his shoulder" is angry because of some thing that happened in the past. It is easy for a person to get in a fight when he has a chip on his shoulder, because he is already angry about something else. To start a fight, men used to put chips of wood on their shoulder and challenge others to "try to knock it off" (о мальчишках-драчунах, носивших щепку на плече и предлагавших сбить её всем желающим вступить в драку).

Example 4: Tom had a tough time growing up, so he's got a bit of a chip on his shoulder.

You can use the definite article ("the") which sounds more general ("a chip on the shoulder"), but more often people use the personal pronoun ("his", "her", "their") to say that that specific person has "a chip on his (her, their) shoulder." (GoEnglish.com)

knock a chip from / off sb's shoulder — принять чей-л. вызов, вступить в драку

See
[лезть на рожон]
[nurse a grudge]
[затаить обиду]