Image reconstruction is a typical ill-posed inverse problem that has attracted increasing attention owing to its extensive use. To cope with the ill-posed nature of this problem, many regularizers have been presented to regularize the reconstruction process. One of the most popular regularizers in the literature is total variation (TV), known for its capability of preserving edges. However, TV-based reconstruction methods often tend to produce staircase-like artifacts since they favor piecewise constant solutions. To overcome this drawback, we propose to develop a second-order total generalized variation (TGVα2)-based image reconstruction model with a combined L1,2 data-fidelity term. The proposed model is applicable for restoration of blurred images with mixed Gaussian-impulse noise, and can be effectively used for undersampled magnetic resonance imaging. To further enhance the image reconstruction, a box constraint is incorporated into the proposed model by simply projecting all pixel values of the reconstructed image to lie in a certain interval (e.g., 0, 1 for normalized images and [0, 255] for 8-bit images). An optimization algorithm based on an alternating direction method of multipliers is developed to solve the proposed box-constrained image reconstruction model. Comprehensive numerical experiments have been conducted to compare our proposed method with some state-of-the-art reconstruction techniques. The experimental results have demonstrated its superior performance in terms of both quantitative evaluation and visual quality.