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Classical Cepheids (DCEPs) are the most important primary indicators of theextragalactic distance scale. Establishing the dependence on metallicity oftheir period--luminosity and period--Wesenheit ($PLZ$/$PWZ$) relations has deepconsequences on the calibration of secondary distance indicators that lead tothe final estimate of the Hubble constant (H$_0$). We collected high-resolutionspectroscopy for 47 DCEPs plus 1 BL Her variables with HARPS-N@TNG and derivedaccurate atmospheric parameters, radial velocities and metal abundances. Wemeasured spectral lines for 29 species and characterized their chemicalabundances, finding very good agreement with previous results. We re-determinedthe ephemerides for the program stars and measured their intensity-averagedmagnitudes in the $V,I,J,H,K_s$ bands.  We complemented our sample with literature data and used the Gaia Early DataRelease 3 (EDR3) to investigate the $PLZ$/$PWZ$ relations for Galactic DCEPs ina variety of filter combinations. We find that the solution without anymetallicity term is ruled out at more than the 5 $\sigma$ level. Our bestestimate for the metallicity dependence of the intercept of the $PLK_s$,$PWJK_s$, $PWVK_s$ and $PWHVI$ relations with three parameters, is$-0.456\pm$0.099, $-0.465\pm$0.071, $-0.459\pm$0.107 and $-0.366\pm$0.089mag/dex, respectively. These values are significantly larger than the recentliterature. The present data are still inconclusive to establish whether or notalso the slope of the relevant relationships depends on metallicity. Applying acorrection to the standard zero point offset of the Gaia parallaxes has thesame effect of reducing by $\sim$22\% the size of the metallicity dependence onthe intercept of the PLZ/PWZ relations.

Cepheid Metallicity in the Leavitt Law (C-MetaLL) survey: I. HARPS-N@TNG spectroscopy of 47 Classical Cepheid and 1 BL Her variables