  We calculate mean values of $\operatorname{GL}_n$-automorphic $L$-functions
twisted by primitive even Dirichlet characters of prime-power conductor, at
arbitrary points within the critical strip, by derivation of special Voronoi
summation formulae. Our calculation is novel in that the twisted sum can be
expressed in terms of the average itself, and also that it sees the derivation
of various new summation formulae in the setting of prime-power modulus. One
consequence, as we explain, is to show the analytic continuation and additive
summation formulae for hyper-Kloosterman Dirichlet series associated to
$\operatorname{GL}_n$-automorphic $L$-functions.
