  We study the variance of sums of the $k$-fold divisor function $d_k(n)$ over
sparse arithmetic progressions, with averaging over both residue classes and
moduli. In a restricted range, we confirm an averaged version of a recent
conjecture about the asymptotics of this variance. This result is closely
related to moments of Dirichlet $L$-functions and our proof relies on the
asymptotic large sieve.
