  The newly observed hidden-charm tetraquark state $Z_{cs}(3985)$, together
with $Z_c(3900)$ and $X(4020)$, are studied in the combined theoretical
framework of the effective range expansion, compositeness relation and the
decay width saturation. The elastic effective-range-expansion approach leads to
sensible results for the scattering lengths, effective ranges and the
compositeness coefficients, $i.e.$, the probabilities to find the
two-charm-meson molecule components in the tetraquark states. The
coupled-channel formalism by including the $J/\psi\pi$ and
$D\bar{D}^*/\bar{D}D^*$ to fulfill the constraints of the compositeness
relation and the decay width, confirms the elastic effective-range-expansion
results for the $Z_c(3900)$, by using the experimental inputs for the ratios of
the decay widths between $D\bar{D}^*/\bar{D}D^*$ and $J/\psi\pi$. With the
results from the elastic effective-range-expansion study as input for the
compositeness, we generalize the discussions to the $Z_{cs}(3985)$ by including
the $J/\psi K^{-}$ and $D_s^{-}D^{*0}/D_s^{*-}D^{0}$, and predict the partial
decay widths of the $J/\psi K^{-}$. Similar calculations are also carried out
for the $X(4020)$ by including the $h_c\pi$ and $D^*\bar{D}^*$, and the partial
decay widths of the $h_c\pi$ is predicted. Our results can provide useful
guidelines for future experimental measurements.
