Divisor problem in arithmetic progressions modulo a prime power

We obtain an asymptotic formula for the average value of the divisor function over the integers $n\le x$ in an arithmetic progression $n\equiv a(\mathrm{m}\mathrm{o}\mathrm{d}q)$, where $q=p^k$ for a prime $p\ge 3$ and a sufficiently large integer $k$. In particular, we break the classical barrier $q\le x^{2\mathrm{/}3}$ for such formulas, and generalise a recent result of R.~Khan (2015), making it uniform in $k$.