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 ) . CNO abundance determinations 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 LTE spectrum synthesis code MOOG , and 1D ATLAS9 atmosphere models to derive stellar abundances , and , where possible , isotopic ^ { 12 } C/ ^ { 13 } C ratios . Results : C , N , Sr , and Ba abundances ( or limits ) are derived for a sample of 27 faint metal-poor stars for which the X-shooter spectra have sufficient S/N ratios . These moderate resolution , low S/N ( \sim 10 - 40 ) spectra prove sufficient to perform limited chemical tagging and enable assignment of these stars into the CEMP sub-classes ( CEMP- s and CEMP-no ) . According to the derived abundances , 17 of our sample stars are CEMP- s and three are CEMP-no , while the remaining seven are carbon-normal . For four CEMP stars , the sub-classification 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 to identify 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 .