Let R be a ring with involution and e, f ∈ R be Hermitian and invertible. We first present some equivalent conditions for paq to be {1, 3 f }-invertible, assuming that p, a, q ∈ R with p ′ pa = a = aqq ′ for some p ′ , q ′ ∈ R and a is {1, 3e}-invertible. Then, these results are applied to give the sufficient and necessary conditions under which Jacobson's lemma and Cline's formula for weighted pseudo core inverses hold. Also, Jacobson's lemma for weighted Moore-Penrose inverses is investigated.