We propose a new Iteratively Reweighted Least Squares (IRLS) algorithm for the problem of completing or denoising low-rank matrices that are structured, e.g., that possess a Hankel, Toeplitz or block-Hankel/Toeplitz structure. The algorithm optimizes an objective based on a non-convex surrogate of the rank by solving a sequence of quadratic problems. Our strategy combines computational efficiency, as it operates on a lower dimensional generator space of the structured matrices, with high statistical accuracy which can be observed in experiments on hard estimation and completion tasks. Our experiments show that the proposed algorithm StrucHMIRLS exhibits an empirical recovery probability close to 1 from fewer samples than the state-of-the-art in a Hankel matrix completion task arising from the problem of spectral super-resolution of badly separated frequencies. Furthermore, we explain how the proposed algorithm for structured low-rank recovery can be used as preprocessing step for improved robustness in frequency or line spectrum estimation problems.