For many centuries, the normal word corresponding to
administration and meaning 'to manage (affairs)' has been
administer (The Rezzoris were minor Austrian gentry administering the outposts of empire—London Review of Books, 1990).
In recent years, however, the longer form
administrate (first recorded in the 17th century) has increasingly been used as a kind of newly invented back-formation, and is now awkwardly challenging
administer in its traditional meanings:
The machinery of such aid is still primed by administrators eager to go out and administrate—Times, 1981
They [speed cameras] are very expensive to install, maintain, administrate and police—Yorkshire Post, 2006.
Administer is, on the other hand, routinely used to mean 'to give (medicine) to a patient'
(I was brimming with alcohol—administered to loosen my tongue—A. Price, 1982)
and is also being increasingly used in two other meanings:
1. to inflict (punishment, blows, etc.) on someone
(Two others held her feet while the headmaster administered the cane—B. Emecheta, 1974).
2. in medical contexts
administer is used instead of
minister to (an injured person, etc.):
The fact that Ranjit is still alive today is a tribute to the ambulance attendants who administered to him at the scene—Oxford Times, 1977
American doctors, being vastly rich, have better things to do with their leisure time than administer to patients at weekends—Times, 1994.