In filtering, control, and other mathematical engineering areas, it is common to use a model-based approach, which splits the problem into two steps: 1) model identification and 2) model-based design. Despite its success, the model-based approach has the shortcoming that the design objective is not taken into account at the identification step, i.e., the model is not optimized for its intended use. This technical note proposes an approach for data-driven filtering and control that combines the identification and the model-based design into one joint problem. The signal of interest is modeled as a missing part of a trajectory of the data generating system. Subsequently, the missing data estimation problem is reformulated as a mosaic-Hankel structured matrix low-rank approximation/completion problem. A local optimization method, based on the variable projections principle, is then used for its numerical solution. The missing data estimation approach and the solution method proposed are illustrated on filtering and smoothing examples.