  We combine a classical idea of Postnikov (1956) with the method of Korobov
(1974) for estimating double Weyl sums, deriving new bounds on short character
sums when the modulus $q$ has a small core $\prod_{p\mid q}p$. Using this
estimate, we improve certain bounds of Gallagher (1972) and Iwaniec (1974) for
the corresponding $L$-functions. In turn, this allows us to improve the error
term in the asymptotic formula for primes in short arithmetic progressions
modulo a power of a fixed prime. As yet another application of our bounds, we
substantially extend the region free of Siegel zeros.
