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 $J/psipi$ and $Dbar{D}^*/bar{D}D^*$ 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 $Dbar{D}^*/bar{D}D^*$ and $J/psipi$. 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 $h_cpi$ and $D^*bar{D}^*$, and the partial decay widths of the $h_cpi$ is predicted. Our results can provide useful guidelines for future experimental measurements.
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