  In this article, we choose the
$[sc]_P[\bar{s}\bar{c}]_A-[sc]_A[\bar{s}\bar{c}]_P$ type tetraquark current to
study the hadronic coupling constants in the strong decays $Y(4660)\to J/\psi
f_0(980)$, $\eta_c \phi(1020)$, $\chi_{c0}\phi(1020)$, $D_s\bar{D}_s$, $ D_s^*
\bar{D}^*_s$, $ D_s \bar{D}^*_s$, $D_s^* \bar{D}_s$, $\psi^\prime \pi^+\pi^-$,
$J/\psi\phi(1020)$ with the QCD sum rules based on solid quark-hadron quality.
The predicted width $\Gamma(Y(4660) )= 74.2^{+29.2}_{-19.2}\,{\rm{MeV}}$ is in
excellent agreement with the experimental data $68\pm 11\pm 1 {\mbox{ MeV}}$
from the Belle collaboration, which supports assigning the $Y(4660)$ to be the
$[sc]_P[\bar{s}\bar{c}]_A-[sc]_A[\bar{s}\bar{c}]_P$ type tetraquark state with
$J^{PC}=1^{--}$. In calculations, we observe that the hadronic coupling
constants $ |G_{Y\psi^\prime f_0}|\gg |G_{Y J/\psi f_0}|$, which is consistent
with the observation of the $Y(4660)$ in the $\psi^\prime\pi^+\pi^-$ mass
spectrum, and favors the $\psi^{\prime}f_0(980)$ molecule assignment. It is
important to search for the process $Y(4660)\to J/\psi \phi(1020)$ to diagnose
the nature of the $Y(4660)$, as the decay is greatly suppressed.
