  Let $P_t$ and $C_t$ be a path and a cycle on $t$ vertices, respectively. In
2021, Choudum {\em et al.} [Disc. Math. 344 (2021) 112244] determined the
structures of $(P_7,C_7,C_4$, diamond)-free and $(P_7,C_7,C_4$, gem)-frees, and
gave correspondingly tight upper bounds to the chromatic numbers of these
graphs. In this paper, we study the structure of $(P_7, C_5$, kite,
paraglider)-free graphs, which is a superfamily of $(P_7, C_5$, diamond)-free
graphs. We show that there is a unique connected imperfect $(P_7, C_5$, kite,
paraglider)-free graph with $\delta(G)\geq\omega(G)+1$, which has no clique
cutsets, no universal cliques, and no pair of vertices of which one's
neighborhoods contains the other's. As a consequence, we show that $(P_7, C_5$,
kite, paraglider)-free graphs are $\chi$-polydet with a binding function
$\omega(G)+1$. Where a {\em diamond} (resp. {\em gem}) consists of a $P_3$
(resp. $P_4$) and a new vertex adjacent to all vertices of the $P_3$ (resp.
$P_4$), a {\em kite} consists of a $P_4$ and a new vertex adjacent to
consecutive three vertices of the $P_4$, and a {\em paraglider} consists of a
$C_4$ and a new vertex adjacent to three vertices of the $C_4$.
