We will introduce the notion of relative (complex) projective structure on a flat (one-parameter) family of smooth complex analytic curves degenerating to a singular nodal curve. Our main concern will be the parametrization of the germs of relative projective structures at a node p of the singular fiber in terms of the germs of the square of the relative dualizing sheaf at p. In contrast to the case of smooth families of curves, there appears an integrability condition for a germ of the latter sheaf to represent a germ of a relative projective structure. As a motivating example, we will also look at degenerations of (meromorphic) projective connections with four regular singularities on the projective line.