This paper focuses on the development of several new Newton-type inequalities for functions whose second derivatives are s-convex in the Breckner sense. By utilizing the setting of Riemann integrals and incorporating a summation parameter p ≥ 1, we derive refined error estimates for Newton-Cotes-like quadrature formulas. The results obtained in this study provide a broad generalization that covers various convexity classes, including P-functions and classical convex functions as special cases. Furthermore, the established inequalities offer potential applications in numerical analysis for improving the precision of quadrature rules.