A correction to the nuclear functional is proposed in order to improve the density of states around the Fermi surface. The induced effect of this correction is to produce a surface-peaked effective mass, whose mean value can be tuned to get closer to 1 for the states close to the Fermi energy. In this work we study the effect of the correction term on global properties of nuclei such as the density of states in $^{40}$Ca and $^{208}$Pb, pairing and specific heat at low temperature in $^{120}$Sn. In the latter application, an explicit temperature-dependent form of the correction term is employed and it is shown that the critical temperature is reduced by 40-60~keV.