  In this paper we develop a Malliavin-Skorohod type calculus for additive
processes in the $L^0$ and $L^1$ settings, extending the probabilistic
interpretation of the Malliavin-Skorohod operators to this context. We prove
calculus rules and obtain a generalization of the Clark-Hausmann-Ocone formula
for random variables in $L^1$. Our theory is then applied to extend the
stochastic integration with respect to volatility modulated L\'evy-driven
Volterra processes recently introduced in the literature. Our work yields to
substantially weaker conditions that permit to cover integration with respect,
e.g. to Volterra processes driven by $\alpha$-stable processes with $\alpha <
2$. The presentation focuses on jump type processes.
