How much influence can any one leader have over their state?

A lot u know Hitler? Yeesh

Quite a bit, yeah, but never "absolute." Even in theoretically absolute monarchies, there are always competing centers of power both inside and outside the government.

Consider the Ming dynasty of China. For a long time, many historians just assumed that the Emperor's power was absolute; the central government issued decrees and the people supposedly "trembled and obeyed." So they studies mainly the documents of the central government and assumed that whatever the Emperor ordered was what actually happened.

In recent decades that's been shown not to be the case. To cite just one example, the historian Sarah Schneewind, in her book "Community Schools and the State in Ming China," shows that although the first Ming emperor ordered the construction of local schools in every county in the empire, very few of those schools were actually built and maintained for any long period of time. Locally powerful people (rich families; Buddhist and Daoist religious institutions; others) resisted the decree because it would have cost them money or the use of land/buildings that they wanted for other purposes.

There's always somebody who can frustrate or oppose the power of an "absolute" monarch to some extent. So in practice no single ruler can have truly absolute power.

Absolute influence, check out North Korea . Imprison or execute the dissidents before they can become dissidents , the North Koreans have major punishments for Lessee Majesty.

Look at the record of Germany in the 1930s and early 1940s under Hitler. Ditto the USSR under Stalin in similar times.

Ditto England under William the Conqueror. Complete change of social and legal conditions for the peasants, compared with conditions under Anglo-Saxon rulers.

Historically, quite a bit.

Absolute.
Not many politicians, Generals or civilians I know of who's want to be stood in front of quad anti-aircraft cannons.

"Kim Jong-Un executes his defence minister with an anti-aircraft gun in front of hundreds of spectators - because he dozed off during a North Korean military ceremony"
http://www.dailymail.co.uk/news/article-3079172/North-Korean-defence-minister-executed-anti-aircraft-fire-disrespecting-leader-Kim-Jong-dozing-military-events-answering-back.html

They can rule with an iron fist. Even be worshipped as "gods" - think of N. Korea, Japan during WWII, Hitler, Stalin.