In this article, we present a new numerical method for effectively solving general second-order ordinary differential equations with mixed boundary conditions. Our approach utilizes a quasi-variable mesh enabling us the flexibility to adapt the mesh density according to different boundary layer problems. By employing a discretization technique that incorporates the construction of exponential spline, we achieve third-order accuracy at internal grid points, while boundary points exhibit fourth-order accuracy. This technique is particularly beneficial while dealing with singular problems since it is based on two half step and one central point and thus it obviates the necessity for any alterations while solving these problems. We provide computational results to demonstrate the efficacy of our proposed method in solving problems. A comparison with recent findings underscores the superior performance of our method.