  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.
