Consider the modular curve $X_0(N)$ over $\mathbb Q$ and for $p\mid N$ consider the reduction modulo $p$ of $X_0(N)$. Let's denote this curve with the symbol $X_p$ (we know that it is a singular curve defined over $\mathbb F_p$).
Now my question is the following: what are the rational points of $X_p$ in terms of the moduli correspondence? To be more precise I would some relation like: a rational point $x\in X_p$ corresponds to an elliptic curve $E$ defined over $\mathbb F_p$ such that...