  The Whittaker module $M_{\psi}$ and its quotient Whittaker module $L_{\psi,
\xi}$ for the twisted affine Nappi-Witten Lie algebra $\widehat{H}_{4}[\tau]$
are studied. For nonsingular type, it is proved that if $\xi\neq 0$, then
$L_{\psi,\xi}$ is irreducible and any irreducible Whittaker
$\widehat{H}_{4}[\tau]$-module of type $\psi$ with ${\bf k}$ acting as a
non-zero scalar $\xi$ is isomorphic to $L_{\psi,\xi}$. Furthermore, for
$\xi=0$, all Whittaker vectors of $L_{\psi, 0}$ are completely determined. For
singular type, the Whittaker vectors of $L_{\psi, \xi}$ with $\xi \neq 0$ are
fully characterized.
