This paper focuses on the robust H control problem for a class of nonlinear switched systems containing neutral uncertainties with average dwell time (ADT) switching. Uncertainties are considered to be nonlinearly dependent on state and state derivative and allowed to appear in channels of state, control input and disturbance input. The robust H control problem of the switched system with stabilisable and unstabilisable subsystems is solvable if the stabilisable and unstabilisable subsystems satisfy certain conditions and admissible switching strategy among them. ADT and piecewise Lyapunov function approaches are applied to achieve the control design. A numerical example is provided to illustrate the effectiveness of the proposed approach.