Let R be a prime ring of characteristic different from two with center Z(R). In the present paper we study the case when a generalized derivation F associated with a nonzero derivation d of R satisfies
$$[F([x, y]_k), [x, y]_k]\in Z(R)$$
for all x, y in some appropriate subset of R, where k is a fixed positive integer. We obtain a description of the structure of R and information on the form of F in terms of the standard identity
and the multiplication by the specific element from the extended centroid.