Posner's second theorem and annihilator conditions with generalized skew derivations

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,...,x_n) 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 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,...,x_n)² is central-valued on R.