  We consider distributions on $\mathbb{R}$ that can be written as the sum of a
non-zero discrete distribution and an absolutely continuous distribution. We
show that such a distribution is quasi-infinitely divisible if and only if its
characteristic function is bounded away from zero, thus giving a new class of
quasi-infinitely divisible distributions. Moreover, for this class of
distributions we characterize the existence of the $g$-moment for certain
functions $g$.
