  Given a Fano variety $Y$ and a simple normal crossings divisor $D\subseteq Y$
which is anti-canonical, we prove a formula relating counts of discs with
boundary on a Lagrangian $L\subseteq Y\backslash D$ to counts of rational
curves in $Y$, under suitable positivity assumptions on $L$. This formula
seriously constrains the topology of $L$ in many examples. Our main application
is a super-potential formula for Fano cyclic coverings $X$ of $Y$. As a
corollary, we show that all the small components of the Fukaya category of a
Fano hypersurface $X\subseteq \mathbb{P}^{n+1}$ are split-generated by monotone
Lagrangian tori.
