  Under the assignment of $D_s(3040)$ as a radially excited p-wave $\bar cs$
state with $J^P=1^+$, we compute the $B_s\to D_s (3040)$ form factors within
the covariant light-front quark model. Two classification schemes for the
p-wave $\bar cs$ meson are adopted. We also use our results to predict the
branching ratios (BRs), polarization fractions and angular asymmetries in
semileptonic $B_s\to D_s(3040)\ell\bar\nu(\ell=e,\mu,\tau)$. The BRs are found
to have the order of $10^{-3}$ for $\ell=e,\mu$ and $10^{-5}$ for $\ell=\tau$.
We find that the polarization fractions and the angular asymmetries could be
useful to pin down the ambiguities of the quantum numbers for $D_s(3040)$. In
addition, we investigate the nonleptonic $B_s\to D_s(3040)M$ decays under the
factorization method, where $M$ denotes a charged pseudoscalar or a vector
meson. The BRs of $B_s\to D_s(3040)\rho$ and $B_s\to D_s(3040)D_s^*$ reach the
order of $10^{-3}$, while the other channels are typically smaller by 1-2
orders.
