Almost universal mixed sums of squares and polygonal numbers

For each integer $m\ge 3$, let $P_m(x)$ denote the generalized $m$-gonal number $\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in \mathbb{Z}$. Given positive integers $a,b,c,k$ and an odd prime number $p$ with $p\nmid c$, we employ the theory of ternary quadratic forms to determine completely when the mixed sum $a{x}^2+b{y}^2+c{P}_{p^k+2}(z)$ represents all but finitely many positive integers.