Let 𝓡 be a prime ring of characteristic different from 2, $𝒬_r$ be its right Martindale quotient ring and 𝓒 be its extended centroid. Suppose that 𝒢 is a non-zero generalized skew derivation of 𝓡 and f(x₁,..., xₙ) is a non-central multilinear polynomial over 𝓒 with n non-commuting variables. If there exists a non-zero element a of 𝓡 such that a[𝒢 (f(r₁,..., rₙ)),f(r₁, ..., rₙ)] = 0 for all r₁, ..., rₙ ∈ 𝓡, then one of the following holds:
(a) there exists λ ∈ 𝓒 such that 𝒢 (x) = λx for all x ∈ 𝓡;
(b) there exist $q ∈ 𝒬_r$ and λ ∈ 𝓒 such that 𝒢 (x) = (q+λ)x + xq for all x ∈ 𝓡 and f(x₁, ..., xₙ)² is central-valued on 𝓡.