Morphisms on the modular curve $X_0(p)$ and degree $6$ points

Let $p$ be a prime. We study non-constant morphisms $f:X_0(p)_\mathbb \to Y$, where $Y/\mathbb Q$ is a curve of genus $\geq 2$. We prove that for $p<3000$ such an $f$ of degree $d>1$ must be isomorphic to the quotient map $X_0(p)\to X_0^+(p)$. Supported by computational and theoretical evidence, we also conjecture that this is true for all primes $p$. These results allow us to classify all points of degree $\leq 25$ on $X_0(p)$ that come from a map to some curve of genus $\geq 2$. As an application, we were able to determine all curves $X_0(p)$ with infinitely many points of degree $6$ over $\mathbb Q$ except for $p=193$, continuing the previous results on small degree points on $X_0(N)$.