  A prime $p$ is called balancing non-Wieferich prime if
$B_{p-(\frac{8}{p})}\not\equiv 0\, (mod \, p^2)$, where $B_n$ be the $n$-th
balancing number and $\displaystyle\bigg(\frac{8}{p}\bigg)$ denotes the Jacobi
symbol. Under the assumption of the $abc$ conjecture for the number field
$\mathbb{Q}[\sqrt{2}]$, S. S. Rout proved that there are at least $O(\log
x/\log\log x)$ said primes $p\equiv 1 \, (mod\, r)$, where $r>2$ be any fixed
integer. In this paper, we improve the lower bound such that for any given
integer $r>2$ there are $\gg\log x$ primes $p\leq x$ satisfies
$B_{p-(\frac{8}{p})}\not\equiv 0\, (mod \, p^2)$ and $p\equiv 1 \, (mod\, r)$,
under the $abc$ conjecture for the number field $\mathbb{Q}[\sqrt{2}]$. This
improves the recent result of Y. Wang and Y. Ding by adding an additional
condition that the primes $p$ are in arithmetic progression.
