Context. An increasing fraction of carbon-enhanced metal-poor (CEMP) stars is found as their iron abundance, [Fe/H], decreases below [Fe/H] = −2.0. The CEMP-s stars have the highest absolute carbon abundances, [C/H], and are thought to owe their enrichment in carbon and the slow neutron-capture (s-process) elements to mass transfer from a former asymptotic giant branch (AGB) binary companion. The most Fe-poor CEMP stars are normally single, exhibit somewhat lower [C/H] than CEMP-s stars, but show no s-process element enhancement (CEMP-no stars). Abundance determinations of CNO offer clues to their formation sites. Aims. Our aim is to use the medium-resolution spectrograph X-Shooter/VLT to determine stellar parameters and abundances for C, N, Sr, and Ba in several classes of CEMP stars in order to further classify and constrain the astrophysical formation sites of these stars. Methods. Atmospheric parameters for our programme stars were estimated from a combination of V − K photometry, model isochrone fits, and estimates from a modified version of the SDSS/SEGUE spectroscopic pipeline. We then used X-Shooter spectra in conjunction with the 1D local thermodynamic equilibrium spectrum synthesis code MOOG, 1D ATLAS9 atmosphere models to derive stellar abundances, and, where possible, isotopic 12 C/ 13 C ratios. Results. Abundances (or limits) of C, N, Sr, and Ba are derived for a sample of 27 faint metal-poor stars for which the X-Shooter spectra have sufficient signal-to-noise ratios (S/N). These moderate resolution, low S/N (∼10−40) spectra prove sufficient to perform limited chemical tagging and enable assignment of these stars into the CEMP subclasses (CEMP-s and CEMP-no). According to the derived abundances, 17 of our sample stars are CEMP-s and 3 are CEMP-no, while the remaining 7 are carbon-normal. For four CEMP stars, the subclassification remains uncertain, and two of them may be pulsating AGB stars. Conclusions. The derived stellar abundances trace the formation processes and sites of our sample stars. The [C/N] abundance ratio is useful for identifying stars with chemical compositions unaffected by internal mixing, and the [Sr/Ba] abundance ratio allows us to distinguish between CEMP-s stars with AGB progenitors and the CEMP-no stars. Suggested formation sites for the latter include faint supernovae with mixing and fallback and/or primordial, rapidly-rotating, massive stars (spinstars). X-Shooter spectra have thus proved to be valuable tools in the continued search for their origin.