  Given a Tychonoff space $X$, let $A(X)$ be the free Abelian topological group
over $X$ in the sense of Markov. For every $n\in\mathbb{N}$, let $A_n(X)$
denote the subspace of $A(X)$ that consists of words of reduced length at most
$n$ with respect to the free basis $X$. In this paper, we show that $A_4(X)$ is
a $k$-space if and only if $A(X)$ is a $k$-space for the non-metrizable
La\v{s}nev space $X$, which gives a complementary for one result of K.
Yamada's. In addition, we also show that, under the assumption of
$\flat=\omega_1$, the subspace $A_3(X)$ is a $k$-space if and only if $A(X)$ is
a $k$-space for the non-metrizable La\v{s}nev space $X$. However, under the
assumption of $\flat>\omega_1$, we provide a non-metrizable La\v{s}nev space
$X$ such that $A_3(X)$ is a $k$-space but $A(X)$ is not a $k$-space.
