Classical Cepheids (DCEPs) are the most important primary indicators of the extragalactic distance scale. Establishing the dependence on metallicity of their period-luminosity and period-Wesenheit (PLZ/PWZ) relations has deep consequences on the calibration of secondary distance indicators that lead to the final estimate of the Hubble constant (H0). We collected high-resolution spectroscopy for 47 DCEPs plus 1 BL Her variables with HARPS-N@TNG and derived accurate atmospheric parameters, radial velocities, and metal abundances. We measured spectral lines for 29 species and characterized their chemical abundances, finding very good agreement with previous results. We re-determined the ephemerides for the program stars and measured their intensity-averaged magnitudes in the V, I, J, H, Ks bands. We complemented our sample with literature data and used the Gaia Early Data Release 3 (EDR3) to investigate the PLZ/PWZ relations for Galactic DCEPs in a variety of filter combinations. We find that the solution without any metallicity term is ruled out at more than the 5{sigma} level. Our best estimate for the metallicity dependence of the intercept of the PLKs, PWJKs, PWVKs, and PWHVI relations with three parameters is -0.456+/-0.099, -0.465+/-0.071, -0.459+/-0.107, and -0.366+/-0.089mag/dex, respectively. These values are significantly larger than the recent literature. The present data are still inconclusive to establish whether or not also the slope of the relevant relationships depends on metallicity. Applying a correction to the standard zero-point offset of the Gaia parallaxes has the same effect of reducing by ~22 per cent the size of the metallicity dependence on the intercept of the PLZ/PWZ relations.