To have a personal issue with someone or a selfish motive behind one's actions.
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To have a private grudge or selfish motive behind what you say.
When someone has an ax to grind, their words come from a hidden aim, not fair judgment.
It is used to question whether someone is really being even-handed.
"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 complaint or selfish motive driving someone's words or actions