The joint problem of reconstruction/feature extraction is a challenging task in image processing. It consists in performing, in a joint manner, the restoration of an image and the extraction of its features. In this work, we firstly propose a novel nonsmooth and nonconvex variational formulation of the problem. For this purpose, we introduce a versatile generalised Gaussian prior whose parameters, including its exponents, are space-variant. Secondly, we design an alternating proximal-based optimisation algorithm that efficiently exploits the structure of the proposed nonconvex objective function. We also analyze the convergence of this algorithm. As shown in numerical experiments conducted on joint segmentation/deblurring tasks, the proposed method provides high-quality results.