The idiom “to have an axe to grind” means:

Concept

An idiom is a fixed expression whose conventional meaning is not normally obtained by taking each word literally.

To interpret an idiom, identify its established figurative sense and test whether that sense makes a natural substitution in context.

Application

The idiom “to have an axe to grind” describes having a private or self-interested motive that influences one’s words or actions. Therefore, “to have some personal interest to serve” expresses the intended meaning.

Contrast

• “To desert” means to abandon someone, a place, a cause, or a duty.

• “To break through reserve or stiffness” refers to making an initially formal or awkward interaction more relaxed.

• “None of these” would apply only if every listed meaning were unsuitable.

Cross-check

In the sentence “He criticised the plan because he had an axe to grind,” the phrase explains a personal motive behind the criticism. This confirms the meaning “to have some personal interest to serve.”