This article is concerned with the mixed H-2/H-infinity control problem over a finite horizon for a class of nonlinear Markovian jump systems with both stochastic nonlinearities and probabilistic sensor failures. The stochastic nonlinearities described by statistical means could cover several types of well-studied nonlinearities, and the failure probability for each sensor is governed by an individual random variable satisfying a certain probability distribution over a given interval. The purpose of the addressed problem is to design state feedback controllers such that the closed-loop system achieves the expected H-2 performance requirement with a guaranteed H-infinity disturbance attenuation level. The solvability of the addressed control problem is expressed as the feasibility of certain coupled matrix equations. The controller gain at each time instant k can be obtained by solving the corresponding set of matrix equations. A numerical example is given to illustrate the effectiveness and applicability of the proposed algorithm.