Many executive department heads are elected in their own right, not appointed by the governor. how can that affect the governor's ability to direct them? why?

If the head of the department is elected by the governor then first, the governor will have an influence over who the head of the department is, and second, it will mean that this person will have to seek approval of the governor - this will support cooperation.

On the other hand, being elected in their own right will support independence.