We study variable-range hopping in the quantum Hall effect regime in the presence of a metallic gate parallel to the plane of a two-dimensional electron gas. Screening of the Coulomb interaction by the gate causes the partial filling of the Coulomb gap in the density of localized states. At low enough temperatures this leads to a substantial enhancement and a different temperature behavior of the hopping conductivity. As a result, the diagonal conductivity peaks become much wider. The power-law dependence of the width of the peaks on the temperature changes: the corresponding exponent turns out to be twice as small as that for gateless structures. The width dependences on the current in the non-Ohmic regime and on the frequency for the absorption of the electromagnetic waves experience a similar modification. The experimental observation of the crossovers predicted may demonstrate the important role of the Coulomb interaction in the integer quantum Hall regime.