  The practice of compressed sensing suffers importantly in terms of the
efficiency/accuracy trade-off when acquiring noisy signals prior to
measurement. It is rather common to find results treating the noise affecting
the measurements, avoiding in this way to face the so-called
$\textit{noise-folding}$ phenomenon, related to the noise in the signal,
eventually amplified by the measurement procedure. In this paper, we present
two new decoding procedures, combining $\ell_1$-minimization followed by either
a regularized selective least $p$-powers or an iterative hard thresholding,
which not only are able to reduce this component of the original noise, but
also have enhanced properties in terms of support identification with respect
to the sole $\ell_1$-minimization or iteratively re-weighted
$\ell_1$-minimization. We prove such features, providing relatively simple and
precise theoretical guarantees. We additionally confirm and support the
theoretical results by extensive numerical simulations, which give a statistics
of the robustness of the new decoding procedures with respect to more classical
$\ell_1$-minimization and iteratively re-weighted $\ell_1$-minimization.
