  In the framework of the improved chromomagnetic interaction model, we
complete a systematic study of the $S$-wave tetraquark states
$Qq\bar{Q}\bar{q}$ ($Q=c,b$, and $q=u,d,s$) with different quantum numbers,
$J^{PC}=0^{+(+)}$, $1^{+(\pm)}$, and $2^{+(+)}$. The mass spectra of tetraquark
states are predicted and the possible decay channels are analyzed by
considering both the angular momentum and $\mathcal{C}$-parity conservation.
The recently observed hidden-charm tetraquark states with strangeness, such as
$Z_{cs}(3985)^-$, $X(3960)$, and $Z_{cs}(4220)^+$, can be well explained in our
model. Besides, based on the wave function of each tetraquark state, we find
that the low-lying states of each $Qq\bar{Q}\bar{q}$ configuration have a large
overlap to the $Q\bar Q$ and $q\bar q$ meson basis, instead of $Q\bar q$ and
$q\bar Q$ meson basis. This indicates one can search these tetraquark states in
future experiments via the channel of $Q\bar Q$ and $q\bar q$ mesons.
