For the pure feedback systems with uncertain actuator nonlinearity and non-differentiable non-affine function, a novel adaptive neural control scheme is proposed. Firstly, the assumption that the non-affine function must be differentiable everywhere with respect to control input has been canceled; in addition, the proposed approach can not only be applicable to actuator input dead zone nonlinearity, but also to backlash nonlinearity without changing the controller. Secondly, the neural network (NN) is used to approximate unknown nonlinear functions of system generated in the process of control design and a nonlinear robust term is introduced to eliminate the actuator nonlinearity modeling error, the NN approximation error and the external disturbances. Semi-globally uniformly ultimately boundedness of all signals in the closed loop system is analytically proved by utilizing Lyapunov theory. Finally, the effectiveness of the designed method is demonstrated via two examples.