This paper investigates the finite-time cluster synchronization problem of nonlinearly coupled and discontinuous Lur'e networks. By introducing the definition of Filippov regularization process, the discontinuously nonlinear function is transformed into a function-valued set. Then, a measurable function is selected form the Filippov set to ensure the existence of the solution for the discontinuous system according to the measurable selection theorem. By designing the finite-time pinning controllers, some sufficient conditions are obtained for cluster synchronization of the discontinuous Lur'e network based on the finite time stability theory. In addition, the settling time for achieving the cluster synchronization is also estimated. And finally, a numerical example is presented to illustrate the validity of theoretical analysis.