If the pair $(X,D)$ is log canonical and $S$ be a component of $D$ with coefficient 1, then we have adjunction type formula as $$K_S+D_S=(K_X+D)|_S$$
If $Z\to X$ be a finite map, then often there exists a divisor $B$ on $X$ s.t, $K_X+B$ is $\mathbb Q$-Cartier and $K_Z=f^*(K_X+B)$.
Note: If $K_X+B$ is not $\mathbb Q$-Cartier, it is not clear what the adjunction formula should mean, but even then one can have a sort of adjunction formula involving $Ext$'s but this is Grothendieck Duality.
See Kawamata-Kodaira canonical bundle formula also