  The possible existence of an eV-mass sterile neutrino, slightly mixing with
ordinary active neutrinos, is not yet excluded by neutrino oscillation
experiments. Assuming neutrinos to be Majorana particles, we explore the impact
of such a sterile neutrino on the effective neutrino mass of neutrinoless
double-beta decays $\langle m \rangle^\prime_{ee} \equiv m^{}_1 |V^{}_{e1}|^2
e^{{\rm i}\rho} + m^{}_2 |V^{}_{e2}|^2 + m^{}_3 |V^{}_{e3}|^2 e^{{\rm i}\sigma}
+ m^{}_4 |V^{}_{e4}|^2 e^{{\rm i}\omega}$, where $m^{}_i$ and $V^{}_{ei}$ (for
$i = 1, 2, 3, 4$) denote respectively the absolute masses and the first-row
elements of the 4$\times$4 neutrino flavor mixing matrix $V$, for which a full
parametrization involves three Majorana-type CP-violating phases $\{\rho,
\sigma, \omega\}$. A zero effective neutrino mass $|\langle m
\rangle^\prime_{ee}| = 0$ is possible no matter whether three active neutrinos
take the normal or inverted mass ordering, and its implications for the
parameter space are examined in great detail. In particular, given the best-fit
values of $m^{}_4 \approx 1.3~{\rm eV}$ and $|V^{}_{e4}|^2 \approx 0.019$ from
the latest global analysis of neutrino oscillation data, a three-dimensional
view of $|\langle m \rangle^\prime_{ee}|$ in the $(m^{}_1, \rho)$-plane is
presented and further compared with that of the counterpart $|\langle m
\rangle^{}_{ee}|$ in the absence of any sterile neutrino.
