Computing the Weil representation of a superelliptic curve

We study the Weil representation $\rho$ of a curve over a $p$-adic field with potential reduction of compact type. We show that $\rho$ can be reconstructed from its stable reduction. For superelliptic curves of the form $y^n=f\left(x\right)$ at primes $p$ whose residue characteristic is prime to the exponent $n$ we make this explicit.