axe

have an axe to grind have a private, sometimes malign, motive for doing or being involved in something.

☞ The expression originated in a story told by the American statesman and scientist Benjamin Franklin and was used first in the USA, especially with reference to politics, but it is now in general use.

2011Financial Times It is not impossible to imagine a scenario … where losses might stir up a row; particularly if political groups had an axe to grind, as in Switzerland.