Anybody can act with purpose; I drive to the grocery store once a week to buy groceries. That was my purpose. I know what it is, I'm not guessing. I decided it myself, or accepted it as my purpose for some hours as assigned by my wife.
"A purpose" is more general, it means an overarching life purpose. A doctor can be a man with a purpose: To heal people.
"A purpose" typically implies something with no defined end. The doctor can heal people as long as he is able. For another person, their purpose may be to entertain people.
It is a subtle difference, but I think in your case you want "a purpose". A sledgehammer is a tool with a purpose: To hammer things hard. It actually has no "purpose" of its own choice or assigned by somebody, it was built to hammer things hard.
While "a man with purpose" implies the man is acting on his own to accomplish some specific finite task, like me shopping for groceries. Shopping for groceries does not define my life, but on that trip I am a man acting with purpose, to acquire groceries for the week.
So if your character is not internally driven to accomplish something they have chosen, but has been created by some entity, then they are more like a sledgehammer: it was created by humans to do something specific. And he was created by some entity to do something specific. He is a man with a purpose; and may not even be sure exactly what that purpose is.