Morphisms on the modular curve X₀(p) and degree $6$ points

Let p be a prime. We study the morphisms f:X₀(p)Q → Y, where Y/ Q is a curve of genus ≥2. We prove that for $p<3000$ such an f must be isomorphic to the quotient map X₀(p)→ X₀⁺(p). This allows us to classify all points of degree ≤ 25 on X₀(p) that come from a map to some curve of genus ≥ 2. As an application, we were able to determine all curves X₀(p) with infinitely many points of degree $6$ over Q, continuing the previous results on small degree points on X₀(N).