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Recent promising effort for spectral reconstruction (SR) focuses on learning a complicated mapping through using a deeper and wider convolutional neural networks (CNNs). Nevertheless, most CNN-based SR algorithms neglect to explore the camera spectral sensitivity (CSS) prior and interdependencies among intermediate features, thus limiting the representation ability of the network and performance of SR. To conquer these issues, we propose a novel adaptive weighted attention network (AWAN) for SR, whose backbone is stacked with multiple dual residual attention blocks (DRAB) decorating with long and short skip connections to form the dual residual learning. Concretely, we investigate an adaptive weighted channel attention (AWCA) module to reallocate channel-wise feature responses via integrating correlations between channels. Furthermore, a patch-level second-order non-local (PSNL) module is developed to capture long-range spatial contextual information by second-order non-local operations for more powerful feature representations. Based on the fact that the recovered RGB images can be projected by the reconstructed hyperspectral image (HSI) and the given CSS function, we incorporate the discrepancies of the RGB images and HSIs as a finer constraint for more accurate reconstruction. Experimental results demonstrate the effectiveness of our proposed AWAN network in terms of quantitative comparison and perceptual quality over other state-of-the-art SR methods. In the NTIRE 2020 Spectral Reconstruction Challenge, our entries obtain the 1st ranking on the Clean track and the 3rd place on the Real World track. Codes are available at https://github.com/Deep-imagelab/AWAN.