We propose a new Iteratively Reweighted Least Squares (IRLS) algorithm for the problem of completing a low-rank matrix that is linearly structured, e.g., that possesses a Hankel, Toeplitz or block-Hankel/Toeplitz structures, which is of relevance for the harmonic retrieval or super-resolution problem. The algorithm optimizes a non-convex surrogate of the rank by minimizing well-chosen quadratic upper bounds of the smoothed surrogate. We establish a quadratic local convergence rate of the developed IRLS strategy if the linear structure is Hankel, with high probability if the provided entries of the matrix are sampled uniformly at random, and if the matrix to be completed fulfills a suitable coherence condition. Our strategy combines computational efficiency, as the dimensionality of its optimization variables scales sub-linearly in the matrix dimensions, with a favorable data efficiency, as it can be observed in experiments on hard completion tasks. In particular, our experiments show that the proposed algorithm exhibits an empirical recovery probability close to one from fewer samples than existing state-of-the-art approaches for the Hankel matrix completion task arising from the problem of spectral super-resolution of frequencies with small separation.