  The Borodin-Kostochka Conjecture states that for a graph $G$, if
$\Delta(G)\geq9$, then $\chi(G)\leq\max\{\Delta(G)-1,\omega(G)\}$. We use $P_t$
and $C_t$ to denote a path and a cycle on $t$ vertices, respectively. Let
$C=v_1v_2v_3v_4v_5v_1$ be an induced $C_5$. A {\em $C_5^+$} is a graph obtained
from $C$ by adding a $C_3=xyzx$ and a $P_2=t_1t_2$ such that (1) $x$ and $y$
are both exactly adjacent to $v_1,v_2,v_3$ in $V(C)$, $z$ is exactly adjacent
to $v_2$ in $V(C)$, $t_1$ is exactly adjacent to $v_4,v_5$ in $V(C)$ and $t_2$
is exactly adjacent to $v_1,v_4,v_5$ in $V(C)$, (2) $t_1$ is exactly adjacent
to $z$ in $\{x,y,z\}$ and $t_2$ has no neighbors in $\{x,y,z\}$. In this paper,
we show that the Borodin-Kostochka Conjecture holds for ($P_6,C_4,H$)-free
graphs, where $H\in \{K_7,C_5^+\}$. This generalizes some results of Gupta and
Pradhan in \cite{GP21,GP24}.
