We search for unresolved X-ray emission from lensed sources in the FOV of 11 CLASH clusters with Chandra data. We consider the solid angle in the lens plane corresponding to a magnification $mu>1.5$, that amounts to a total of ~100 arcmin$^2$. Our main goal is to assess the efficiency of massive clusters as cosmic telescopes to explore the faint end of X-ray extragalactic source population. We search for X-ray emission from strongly lensed sources identified in the optical, and perform an untargeted detection of lensed X-ray sources. We detect X-ray emission only in 9 out of 849 lensed/background optical sources. The stacked emission of the sources without detection does not reveal any signal in any band. Based on the untargeted detection, we find 66 additional X-ray sources that are consistent with being lensed sources. After accounting for completeness and sky coverage, we measure for the first time the soft- and hard-band number counts of lensed X-ray sources. The results are consistent with current modelization of the AGN population distribution. The distribution of de-lensed fluxes of the sources identified in moderately deep CLASH fields reaches a flux limit of ~$10^{-16}$ and ~$10^{-15}$ erg/s/cm$^{2}$ in the soft and hard bands, respectively. We conclude that, in order to match the depth of the CDFS exploiting massive clusters as cosmic telescopes, the required number of cluster fields is about two orders of magnitude larger than that offered by the 20 years Chandra archive. A significant step forward will be made when future X-ray facilities, with ~1 angular resolution and large effective area, will allow the serendipitous discovery of rare, strongly lensed high-$z$ X-ray sources, enabling the study of faint AGN activity in early Universe and the measurement of gravitational time delays in the X-ray variability of multiply imaged AGN.
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