Choice C is correct. It's given that line $r$ is perpendicular to line $p$ in the $xy$-plane. This means that the slope of line $r$ is the negative reciprocal of the slope of line $p$. If the equation for line $p$ is rewritten in slope-intercept form $y=mx+b$, where $m$ and $b$ are constants, then $m$ is the slope of the line and $\left(0, b\right)$ is its $y$-intercept. Subtracting $18x$ from both sides of the equation $2y+18x=9$ yields $2y=-18x+9$. Dividing both sides of this equation by $2$ yields $y=-9x+\dfrac{9}{2}$. It follows that the slope of line $p$ is $-9$. The negative reciprocal of a number is $-1$ divided by the number. Therefore, the negative reciprocal of $-9$ is $\dfrac{-1}{-9}$ or $\dfrac{1}{9}$. Thus, the slope of line $r$ is $\dfrac{1}{9}$.

Choice $A$ is incorrect. This is the slope of line $p$, not line $r$.

Choice $B$ is incorrect. This is the reciprocal, not the negative reciprocal, of the slope of line $p$.

Choice $D$ is incorrect. This is the negative, not the negative reciprocal, of the slope of line $p$.