A Morley-Wang-Xu (MWX) element method with a simply modified right hand side is proposed for a fourth order elliptic singular perturbation problem, in which the discrete bilinear form is standard as usual nonconforming finite element methods. The sharp error analysis is given for this MWX element method. And the Nitsche's technique is applied to the MXW element method to achieve the optimal convergence rate in the case of the boundary layers. An important feature of the MWX element method is solver-friendly. Based on a discrete Stokes complex in two dimensions, the MWX element method is decoupled into one Lagrange element method of Poisson equation, two Morley element methods of Poisson equation and one nonconforming $P_1$-$P_0$ element method of Brinkman problem, which implies efficient and robust solvers for the MWX element method. Some numerical examples are provided to verify the theoretical results.