On projections of smooth and nodal plane curves

Suppose that $C\subset\mathbb P^2$ is a general enough nodal plane curve of degree $>2$, $\nu\colon\hat C\to C$ is its normalization, and $\pi\colon C'\to\mathbb P^1$ is a finite morphism simply ramified over the same set of points as a projection $\mathrm{pr}_p\circ\nu\colon\hat C \to\mathbb P^1$, where $p\in\mathbb P^2\setminus C$ (if $\deg C=3$, one should assume in addition that $\deg\pi\ne4$). We prove that the morphism $\pi$ is equivalent to such a projection if and only if it extends to a finite morphism $X\to(\mathbb P^2)^*$ ramified over $C^*$, where $X$ is a smooth surface.
As a by-product, we prove the Chisini conjecture for mappings ramified over duals to general nodal curves of any degree $\ge3$ except for duals to smooth cubics; this strengthens one of Victor Kulikov's results.