When is a state administrator said to have a measure of independence from the governor?
How much independence a state administrator has from the governor who has appointed or nominated him or her to that office is dependent upon the individual state's constitution or upon state laws passed over time. In states where state administrators are directly appointed by the governor they will ostensibly serve, they are often close politically to the governor and view as their primary responsibility the advancement of the governor's agenda. In states, for example, New York and California, the governor nominates an individual to the position, but the state senate must confirm the nominee before he or she can serve in an official capacity. This is an important distinction, as it allows the state legislature to inject its perspective into the process, although, once confirmed, there is no question where the loyalties of the state administrator lie. In short, this is a position that exists to execute the agenda of the governor. The governor appoints or nominates the individual to the position, and there is an expectation that the governor, as the state's highest-ranking elected official, will be the principal superior to the administrator.
A state administrator is a member of the executive branch of that state's government. As such, individuals who serve in that capacity are only as independent as their willingness to risk their positions to defy the instructions of the state's chief executive. State legislatures can take measures to oversee the office as part of their routine oversight responsibilities, including their control of budgetary matters, but the administrator serves largely at the pleasure of the governor.
Now, all of the above is specific to "state administrators" as that term is often employed. There is also the application of that term in reference to state-level officials who oversee the states' securities industries. In such cases, the chief administrator is again nominated and appointed, but often enjoys a modicum of independence from the executive and legislative branches that the more mundane administrative officials discussed above usually do not enjoy.