Sums of von Mangoldt convolutions via trace-sum averages over $U(N)$

We study piecewise polynomial functions ${\delta }_k(c)$ that appear in the conjectured asymptotics for the variance of short-interval sums of von Mangoldt convolutions ${\mathrm{\Lambda }}^{\ast k}$. By a theorem of Kuperberg and Lalín, a unitary matrix integral involving a sum of trace products governs the large-$q$ limit of the function-field analogue of this variance. We identify ${\delta }_k(c)$ as the leading-order coefficient of this integral when the total trace degree and matrix size grow with a fixed ratio $c$. Through an asymptotic analysis of two lattice-point representations for the matrix integral, we obtain explicit forms of ${\delta }_k(c)$. A finite spline expansion and a contour-integral representation expressing ${\delta }_k(c)$ as the inverse Laplace transform of a two-weight block Hankel determinant are also obtained.