The discrete gap breathers (DGBs) in a one-dimensional diatomic chain with K2-K3-K4 potential are analysed.Using the local anharmonicity approximation, the analytical investigation has been implemented. The dependence of the central amplitude of the discrete gap breathers on the breather frequency and the localization parameter are calculated.With increasing breather frequency, the DGB amplitudes decrease. As a function of the localization parameter, the central amplitude exhibits bistability, corresponding to the two branches of the curve ω = ω(ζ). With a nonzero cubic term, the HS mode of DGB profiles becomes weaker. With increasing K3, the HS mode of DGB profiles becomes weaker and a bit narrower. For the LS mode, with increasing Ks, the central particle amplitude becomes larger, and the DGB profile becomes much sharper. But, as k3 increases further, the central particle amplitude of the LS mode becomes smaller.