ax to grind

have (got) an ax to grind {US} and have (got) an axe to grind {UK, US} (have a secret purpose or ulterior motive; || have a disagreement to settle with someone; have a grudge) — преследовать личные или корыстные интересы; преследовать своекорыстные цели; иметь личную заинтересованность; быть лицом заинтересованным; || иметь претензии к кому-л.; ~ иметь зуб на кого-л

Example 1: I don’t think he really likes her. I think he has an axe to grind and that’s why he’s dating her. — Не думаю, что она ему нравится. По-моему, он имеет какой-то корыстный интерес, пожтому и встречается с ней.

Example 2: ...

Another meaning of "axe to grind" (2) is to have a disagreement to settle with someone. If you have an ax to grind you have a grudge — something has been bothering you and you need to tell others about it. (иметь зуб, иметь претензии к кому-л.)

Example: He had an ax to grind with cat owners so he targeted his virus for all websites that contained the word cat. (David W. Miller )

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There’s an old story that Ben Franklin, one of the founders of the United States, had a grindstone, a rock that was used for sharpening blades. One day a man came to Franklin and asked him how the grindstone worked. In order to explain how it worked, Franklin sharpened the man’s axe. Now, do you think the man really wanted to know how the stone worked, or do you just think he wanted his axe sharpened? (English Baby)