Density regularity of $\{x,x+y,xy\}$ in the integers

Fix $s\in \mathbb{N}$. We prove that there exists a subadditive density on $\mathbb{N}$ such that, for every polynomial $P\in \mathbb{Z}[y]$ with $P(0)=0$, every set of positive density contains configurations $\{x,x+P(y),x{y}^s\}$ for arbitrarily large $x>y\ge 2$. When $s=1$, this strengthens Moreira’s partition-regularity result for $\{x,x+P(y),xy\}$ to a density theorem, and when $s>1$ this yields new partition-regularity results.