  In the present paper, we investigate the kaon twist-3 distribution amplitudes
(DAs) $\phi_{p,\sigma}^K$ within the QCD background field approach. The
$SU_f(3)$-breaking effects are studied in detail under a systematical way,
especially the sum rules for the moments of $\phi_{p,\sigma}^K$ are obtained by
keeping all the mass terms in the $s$-quark propagator consistently. After
adding all the uncertainties in quadrature, the first two Gegenbauler moments
of $\phi_{p,\sigma}^K$ are $a^1_{K,p}(1 {\rm GeV}) = -0.376^{+0.103}_{-0.148}$,
$a^2_{K,p}(1 {\rm GeV}) = 0.701^{+0.481}_{-0.491}$, $a^1_{K,\sigma}(1 {\rm
GeV}) = -0.160^{+0.051}_{-0.074}$ and $a^2_{K,\sigma}(1 {\rm GeV}) =
0.369^{+0.163}_{-0.149}$, respectively. Their normalization parameters $\mu_K^p
|_{1\rm GeV} = 1.188^{+0.039}_{-0.043}$ GeV and $\mu_K^\sigma |_{1\rm GeV} =
1.021^{+0.036}_{-0.055}$ GeV. A detailed discussion on the properties of
$\phi^K_{p,\sigma}$ moments shows that the higher-order $s$-quark mass terms
can indeed provide sizable contributions. Furthermore, based on the newly
obtained moments, a model for the kaon twist-3 wavefunction
$\Psi_{p,\sigma}^K(x,\mathbf{k}_\perp)$ with a better end-point behavior is
constructed, which shall be useful for perturbative QCD calculations. As a
byproduct, we make a discussion on the properties of the pion twist-3 DAs.
