  We study a model of multi-excited random walk with non-nearest neighbour
steps on $\mathbb Z$, in which the walk can jump from a vertex $x$ to either
$x+1$ or $x-i$ with $i\in \{1,2,\dots,L\}$, $L\ge 1$. We first point out the
multi-type branching structure of this random walk and then prove a limit
theorem for a related multi-type Galton-Watson process with emigration, which
is of independent interest. Combining this result and the method introduced by
Basdevant and Singh [Probab. Theory Related Fields (2008), 141 (3-4)], we
extend their result (w.r.t the case $L=1$) to our model. More specifically, we
show that in the regime of transience to the right, the walk has positive speed
if and only if the expected total drift $\delta>2$. This confirms a special
case of a conjecture proposed by Davis and Peterson.
