A promising modification of the gyrotron cavity with a little bit wider and fairly long insertion (drift section) into the regular cylindrical part has been studied, both analytically and numerically, by means of both solution of the well-known nonuniform string equation and direct solution of Maxwell equations based on the CST Microwave Studio code. It has been shown that relatively low ohmic losses and an axial field structure that is favorable for the achievement of a high gyrotron efficiency can, in principle, be provided in such a sectioned cavity in the cases of transparent and strongly reflecting drift sections. The decrease in ohmic losses is stronger for the second case because of the smaller amplitude of the wave with a high group velocity in the drift section, but the separation between the operating mode and the closest spurious axial mode is larger for the first case.