The very high fraction of C-enriched stars among the low end of the Galactic-halo metallicity distribution, along with the wide variety in content of their neutron-capture elements, make the understanding of their formation scenario(s) of extreme interest. The combination of accurate determinations of their chemical composition with long-term radial-velocity monitoring is crucial to address this issue. We discuss the first results of long-term radial-velocity monitoring for C-enhanced, metal-poor stars and explore the connection with chemical peculiarities; the data suggest that belonging to a close binary affects nucleosynthesis on the Asymptotic Giant Branch. We also present abundances for the poorly studied element fluorine in a sample of 10 C-enhanced stars with metallicities ranging from [Fe/H] = –1.3 to –2.9 and compare our measurements to the predictions of state-of-the-art nucleosynthesis models. We show how the observed abundances of fluorine in CEMP stars are considerable lower than expected on the basis of the current state-of-the-art theoretical calculations, hinting at our poor understanding of the nucleosynthetic processes that produce this element.