The title is self-explanatory. Is there a word for the desire to have power, or should I just use the phrase?
You can describe someone as being power-hungry:
adjective: Having a strong desire for power.
‘the power-hungry and evil dictator’
You may use powermad. Here's an example from The Empowered Leader by C Miller:
I suspect that most powermad leaders never define themselves that way. Their erosion to the abusive use of power was so gradual that they may not have seen it.
and one from a biography of Theodore Roosevelt by L L Gould:
To a significant portion of his party, moreover, [Theodore] Roosevelt seemed a powermad radical lusting for a return to the highest office in the land.
Thirst for power is an idiomatic expression that is often used to convey the idea.
Thirst for power is also a book by Leon Trotsky
Hitler's Thirst For Power Essay Examples
From Invisible Power:
- Pigott was described as "an iron-fisted decision-maker who cannot satisfy a thirst for power."The NCC chief reacted like a bemused parent, retorting that Ottawa is not a playground merely for the residents, ......
McClelland defines nPow as a desire for power either to control other people (for [one's] own goals) or to achieve higher goals (for the greater good), and describes people high in this trait as seeking neither recognition nor approval from others, but only agreement and compliance. Prsons with a high nPow score are likely to seek power over others.
So someone who desires power can be said to have a high nPow score.