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 non-smooth and non-convex variational formulation of the problem. For this purpose, we introduce a versatile generalised Gaussian prior whose parameters, including its exponent, are space-variant. Secondly, we design an alternating proximal-based optimisation algorithm that efficiently exploits the structure of the proposed non-convex objective function. We also analyse the convergence of this algorithm. As shown in numerical experiments conducted on joint deblurring/segmentation tasks, the proposed method provides high-quality results.