Recovery of arbitrarily positioned samples that are missing in sparse signals recently attracted significant research interest. Sparse signals with heavily corrupted arbitrary positioned samples could be analyzed in the same way as compressive sensed signals by omitting the corrupted samples and considering them as unavailable during the recovery process. The reconstruction of missing samples is done by using one of the well known reconstruction algorithms. In this paper we will propose a very simple and efficient adaptive variable step algorithm, applied directly to the concentration measures, without reformulating the reconstruction problem within the standard linear programming form. Direct application of the gradient approach to the nondifferentiable forms of measures lead us to introduce a variable step size algorithm. A criterion for changing adaptive algorithm parameters is presented. The results are illustrated on the examples with sparse signals, including approximately sparse signals and noisy sparse signals.