Aims. We introduce a novel approach to reconstructing dark matter mass maps from weak gravitational lensing measurements. The cornerstone of the proposed method lies in a new modelling of the matter density field in the Universe as a mixture of two components: (1) a sparsity-based component that captures the non-Gaussian structure of the field, such as peaks or halos at different spatial scales, and (2) a Gaussian random field, which is known to represent the linear characteristics of the field well.Methods. We propose an algorithm called MCALens that jointly estimates these two components. MCALens is based on an alternating minimisation incorporating both sparse recovery and a proximal iterative Wiener filtering.Results. Experimental results on simulated data show that the proposed method exhibits improved estimation accuracy compared to customised mass-map reconstruction methods.Key words: cosmology: observations / techniques: image processing / methods: data analysis / gravitational lensing: weak