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We present a model for non-blind image deconvolution that incorporates the classic iterative method into a deep learning application. Instead of using large over-parameterised generative networks to create sharp picture representations, we build our network based on the iterative Landweber deconvolution algorithm, which is integrated with trainable convolutional layers to enhance the recovered image structures and details. Additional to the data fidelity term, we also add Hessian and sparse constraints as regularization terms to improve the image reconstruction quality. Our proposed model is \textit{self-supervised} and converges to a solution based purely on the input blurred image and respective blur kernel without the requirement of any pre-training. We evaluate our technique using standard computer vision benchmarking datasets as well as real microscope images obtained by our enhanced depth-of-field (EDOF) underwater microscope, demonstrating the capabilities of our model in a real-world application. The quantitative results demonstrate that our approach is competitive with state-of-the-art non-blind image deblurring methods despite having a fraction of the parameters and not being pre-trained, demonstrating the efficiency and efficacy of embedding a classic deconvolution approach inside a deep network.