Aims. We study the effects of non-local thermodynamic equilibrium (NLTE) on the determination of stellar parameters and abundances of Fe, Mg, and Ti from the medium-resolution spectra of FGK stars. Methods. We extended the Payne fitting approach to draw on NLTE and LTE spectral models. These were used to analyse the spectra of the Gaia-ESO benchmark stars and the spectra of 742 stars in 13 open and globular clusters in the Milky Way: NGC 3532, NGC 5927, NGC 2243, NGC 104, NGC 1851, NGC 2808, NGC 362, M 2, NGC 6752, NGC 1904, NGC 4833, NGC 4372, and M15. Results. Our approach accurately recovers effective temperatures, surface gravities, and abundances of the benchmark stars and clusters members. The differences between NLTE and LTE are significant in the metal-poor regime, [Fe/H] ≲ −1. The NLTE [Fe/H] values are systematically higher, whereas the average NLTE [Mg/Fe] abundance ratios are ∼0.15 dex lower, compared to LTE. Our LTE measurements of metallicities and abundances of stars in Galactic clusters are in a good agreement with the literature. Though, for most clusters, our study yields the first estimates of NLTE abundances of Fe, Mg, and Ti. Conclusion. All clusters investigated in this work are homogeneous in Fe and Ti, with the intra-cluster abundance variations of less then 0.04 dex. NGC 2808, NGC 4833, M 2, and M 15 show significant dispersions in [Mg/Fe]. Contrary to common assumptions, the NLTE analysis changes the mean abundance ratios in the clusters, but it does not influence the intra-cluster abundance dispersions.