  We propose and study the weak convergence of a projective splitting algorithm
for solving multi-term composite monotone inclusion problems involving the
finite sum of $n$ maximal monotone operators, each of which having an inner
four-block structure: sum of maximal monotone, Lipschitz continuous, cocoercive
and smooth differentiable operators. We show how to perform backward and
half-forward steps with respect to the maximal monotone and
Lipschitz$+$cocoercive components, respectively, while performing
proximal-Newton steps with respect to smooth differentiable blocks.
