We extend the study of [B. Bouchard, G. Loeper, and Y. Zou, SIAM J. Control Optim., 55 (2017), pp. 3319-3348; G. Loeper, Ann. Appl. Probab., 28 (2018), pp. 2664-2726] stochastic target problems with general market impacts. Namely, we consider a general abstract model which can be associated to a fully nonlinear parabolic equation. Unlike the earlier articles, the equation is not concave, and the regularization/verification approach of our 2017 cannot be applied. We also relax the gamma constraint of the 2017 article. Instead, we need to generalize the a priori estimates of Loeper's article and exhibit smooth solutions from the classical parabolic equations theory. Up to an additional approximating argument, this allows us to show that the superhedging price solves the parabolic equation and that a perfect hedging strategy can be constructed when the coefficients are smooth enough. This representation leads to a general dual formulation. We finally provide an asymptotic expansion around a model without impact.