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We study the Generalized Fermat Equation $x^2 + y^3 = z^p$, to be solved incoprime integers, where $p \ge 7$ is prime. Using modularity and level loweringtechniques, the problem can be reduced to the determination of the sets ofrational points satisfying certain 2-adic and 3-adic conditions on a finite setof twists of the modular curve $X(p)$.  We first develop new local criteria to decide if two elliptic curves withcertain types of potentially good reduction at 2 and 3 can have symplecticallyor anti-symplectically isomorphic $p$-torsion modules. Using these criteria weproduce the minimal list of twists of $X(p)$ that have to be considered, basedon local information at 2 and 3; this list depends on $p \bmod 24$. Recentresults on mod $p$ representations with image in the normalizer of a splitCartan subgroup allow us to reduce the list further in some cases.  Our second main result is the complete solution of the equation when $p =11$, which previously was the smallest unresolved $p$. One relevant newingredient is the use of the `Selmer group Chabauty' method introduced by thethird author in recent work, applied in an Elliptic Curve Chabauty context, todetermine relevant points on $X_0(11)$ defined over certain number fields ofdegree 12. This result is conditional on GRH, which is needed to showcorrectness of the computation of the class groups of five specific numberfields of degree 36.  We also give some partial results for the case $p = 13$.

The generalized Fermat equation with exponents 2, 3, n