  We prove a $q$-analogue of the Carter-Payne theorem for the two special cases
corresponding to moving an arbitrary number of nodes between adjacent rows, or
moving one node between an arbitrary number of rows. As a consequence, we show
that these homomorphism spaces are one dimensional when $q \neq -1$. We apply
these results to complete the classification of the reducible Specht modules
for the Hecke algebras of the symmetric groups when $q \neq-1$. Our methods can
also be used to determine certain other pairs of Specht modules between which
there is a homomorphism. In particular, we describe the homomorphism space from
the trivial module to an arbitrary Specht module.
