Using a high-resolution spectrum of the secondary star in the neutron star binary Cen X-4, we have derived the stellar parameters and veiling caused by the accretion disk in a consistent way. We have used a χ2 minimization procedure to explore a grid of 1,500,000 LTE synthetic spectra computed for a plausible range of both stellar and veiling parameters. Adopting the best model parameters found, we have determined atmospheric abundances of Fe, Ca, Ti, Ni, and Al. These element abundances are supersolar ([Fe/H]=0.23+/-0.10), but only the abundances of Ti and Ni appear to be moderately enhanced (>=1 σ) as compared with the average values of stars of similar iron content. These element abundances can be explained if the secondary star captured a significant amount of matter ejected from a spherically symmetric supernova (SN) explosion of a 4 Msolar He core progenitor and if we assume solar abundances as primordial abundances in the secondary star. The kinematic properties of the system indicate that the neutron star received a natal kick velocity through an aspherical SN and/or an asymmetric neutrino emission. The former scenario might be ruled out, since our model computations cannot produce acceptable fits to the observed abundances. We have also examined whether this system could have formed in the Galactic halo, and our simulations show that this possibility seems unlikely. We also report a new determination of the Li abundance, consistent with previous studies, that is unusually high and close to the cosmic Li abundance in the Galactic disk. Based on observations obtained with the UVES at the VLT Kueyen 8.2 m telescope in program 65.H-0447.