For C a finite tensor category we consider four versions of the central monad, A1,…,A4 on C. Two of them are Hopf monads, and for C pivotal, so are the remaining two. In that case all Ai are isomorphic as Hopf monads. We define a monadic cointegral for Ai to be an Ai-module morphism 1→Ai(D), where D is the distinguished invertible object of C. We relate monadic cointegrals to the categorical cointegral introduced by Shimizu (2019), and, in case C is braided, to an integral for the braided Hopf algebra L=∫XX∨⊗X in C studied by Lyubashenko (1995). Our main motivation stems from the application to finite dimensional quasi-Hopf algebras H. For the category of finite-dimensional H-modules, we relate the four monadic cointegrals (two of which require H to be pivotal) to four existing notions of cointegrals for quasi-Hopf algebras: the usual left/right cointegrals of Hausser and Nill (1994), as well as so-called γ-symmetrised cointegrals in the pivotal case, for γ the modulus of H. For (not necessarily semisimple) modular tensor categories C, Lyubashenko gave actions of surface mapping class groups on certain Hom-spaces of C, in particular of SL(2,Z) on C(L,1). In the case of a factorisable ribbon quasi-Hopf algebra, we give a simple expression for the action of S and T which uses the monadic cointegral.