In this paper, we study the orthogonal polynomials with respect to a singularly perturbed Pollaczek–Jacobi type weight (Formula presented.) By using the ladder operator approach, we establish the second-order difference equations satisfied by the recurrence coefficient (Formula presented.) and the sub-leading coefficient (Formula presented.) of the monic orthogonal polynomials, respectively. We show that the logarithmic derivative of (Formula presented.) can be expressed in terms of a particular Painlevé V transcendent. The large (Formula presented.) asymptotic expansions of (Formula presented.) and (Formula presented.) are obtained by using Dyson's Coulomb fluid method together with the related difference equations. Furthermore, we study the associated Hankel determinant (Formula presented.) and show that a quantity (Formula presented.), allied to the logarithmic derivative of (Formula presented.), can be expressed in terms of the (Formula presented.) -function of a particular Painlevé V. The second-order differential and difference equations for (Formula presented.) are also obtained. In the end, we derive the large (Formula presented.) asymptotics of (Formula presented.) and (Formula presented.) from their relations with (Formula presented.) and (Formula presented.).