We prove the existence of a compactly supported, continuous (except at finitely many points) function gI,m:[0,∞)⟶R for all monomial prime ideals I of R of height one where (R,m) is the homogeneous coordinate ring associated to a projectively normal toric pair (X,D), such that ∫0∞gI,m(λ)dλ=β(I,m), where β(I,m) is the second coefficient of the Hilbert-Kunz function of I with respect to the maximal ideal m, as proved by Huneke-McDermott-Monsky [8]. Using the above result, for standard graded normal affine monoid rings we give a complete description of the class map τm:Cl(R)⟶R introduced in [8] to prove the existence of the second coefficient of the Hilbert-Kunz function. Moreover, we show the function gI,m is multiplicative on Segre products with the expression involving the first two coefficients of the Hilbert plolynomial of the rings and the ideals. © 2021 Elsevier B.V.