Non-Gaussian residual error and noise are common in the real applications, and they can be efficiently addressed by some non-quadratic fidelity terms in the classic variational method. However, they have not been well integrated into the architectures design in the convolutional neural networks (CNN) based image denoising method. In this paper, we propose a deep learning approach to handle non-Gaussian residual error. Our method is developed on an universal approximation property for the probability density functions of the non-Gaussian error/noise. By considering the duality of the maximum likelihood estimation for the non-Gaussian error, an adaptive weighting strategy can be derived for image fidelity. To get a good image prior, a learnable regularizer is adopted. Solving such a problem iteratively can be unrolled as a weighted residual CNN architecture. The main advantage of our method is that the weighted residual block can well handle the non-Gaussian residual, especially for the noise with non-uniformly spatial distribution. Numerical results show that it has better performance on non-Gaussian noise (e.g. Gaussian mixture, random-valued impulse noise) removal than the related existing methods.