In your example "poleaxing" is a typical journalistic pun that works because it is about a politician hoping to be elected at the polls.
Pole - homonym - poll as in elections
to axe = to be cut down with an axe.
A poleaxe is an ancient military weapon*. It also appears in the phrase "He went down like a poleaxed ox" = He collapsed like an ox collapses when hit with a poleaxe.
"heading for a poleaxing" will be slaughtered/defeated heavily at the polls.
*OED
Poleaxe: (n.)
- (a) Originally: a weapon for use in close combat, existing in various forms but generally having a head consisting of an axe blade or hammer-head, balanced at the rear by a pointed fluke. Later also: any of various long-handled weapons of this type, as carried (now only ceremonially) by the bodyguard of a monarch or great personage.)
Verb
1. transitive. To fell, kill, or stun (an animal) with a blow from a poleaxe.
The etymology of poleaxe is obscure but a fair bet is that the "Pole" part of it refers to the Old English "pol" meaning "head" - i.e. "designed to split open a head."