A monad $G=\left(G,\mu ,\eta \right)$ on an additive category $A$ is additive if its underlying endofunctor $G:A\to A$ is an additive functor. One defines an additive comonad in the same vein.
Note that every additive category is Ab-enriched, and an additive monad is then the same as an $\mathrm{Ab}$-enriched monad?.