An attractive proposal to explain the existence and abundances of certain proton-rich isotopes in the cosmos is called the $\nu p$-process, which can operate in neutrino-driven outflows in a core-collapse supernova environment. Here, we show that a realistic modelling of core-collapse supernova hydrodynamics is important for evaluating the $\nu p$-process prospects. This sensitivity arises as a result of the near-critical nature of these outflows, implying that the outflows can have either a subsonic or a supersonic nature, depending on the exact physical conditions near the protoneutron star. Whether or not a robust $\nu p$-process takes place depends on the entropy and expansion timescale of the neutrino-driven outflows.