  In this paper, we study a Ramsey-type problem for equations of the form
$ax+by=p(z)$. We show that if certain technical assumptions hold, then any
2-colouring of the positive integers admits infinitely many monochromatic
solutions to the equation $ax+by=p(z)$. This entails the $2$-Ramseyness of
several notable cases such as the equation $ax+y=z^n$ for arbitrary
$a\in\mathbb{Z}^{+}$ and $n\ge 2$, and also of
$ax+by=a_Dz^D+\dots+a_1z\in\mathbb{Z}[z]$ such that $\text{gcd}(a,b)=1$, $D\ge
2$, $a,b,a_D>0$ and $a_1\neq0$.
