To have a personal issue with someone or a selfish motive behind one's actions.
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To have a personal grudge or selfish reason behind what you are saying or doing.
When someone has an ax to grind, their actions or opinions are motivated by a hidden agenda or resentment, not by fair judgment.
It is used to question whether someone's motives are truly objective.
"Have a Bone to Pick"
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To have a specific complaint you want to raise directly with someone
"A Hidden Agenda"
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A concealed intention that is different from the stated reason
"To Have an Ax to Grind"
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A personal grievance or selfish motive driving someone's words or actions