Completing a data tensor with structured missing components is a challenging task where the missing components are not distributed randomly but they admit some regular patterns, e.g. missing columns and rows or missing blocks/patches. Many of the existing tensor completion algorithms are not able to handle such scenarios. In this paper, we propose a novel and efficient approach for matrix/tensor completion by applying Hankelization and distributed tensor ring decomposition. Our main idea is first Hankelizing an incomplete data tensor in order to obtain high-order tensors and then completing the data tensor by imposing sparse representation on the core tensors in tensor ring format. We apply an efficient over-complete discrete cosine transform dictionary and sparse representation techniques to learn core tensors. Alternating direction methods of multiplier and accelerated proximal gradient approaches are used to solve the underlying optimization problems. Extensive simulations performed on image, video completions and time series forecasting show the validity and applicability of the method for different kinds of structured and random missing elements.