An Inequality with Prime Numbers

Question

Let $m\geq 8$ be an integer, and let $q$ be the smallest prime that is greater than $m$. Let $p$ be a prime that satisfies $q\leq p<q^2$, and let $p_0$ be the smallest prime such that $p_0q>p^2$. I would like to show that $(p_0-m)(q-m)\leq p^2-pm$. I have tried some small examples, and the left-hand side seems to be much less than the right-hand side. However, I cannot seem to find a way to prove the inequality in general. Any advice would be greatly appreciated. Thank you.