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
$$s_4$$
and the multiplication by the specific element from the extended centroid.