  We study Pfaffians that appear in non-perturbative superpotential terms
arising from worldsheet instantons in heterotic theories. A result by Beasley
and Witten shows that these instanton contributions cancel among curves within
a given homology class for Calabi-Yau manifolds that can be described as
hypersurfaces or complete intersections in projective or toric ambient spaces.
We provide a prescription that identifies all $\mathbb{P}^1$ curves in certain
homology classes of complete intersection Calabi-Yau manifolds in products of
projective spaces (CICYs) and cross-check our results by a comparison with the
genus zero Gromov-Witten invariants. We then use this construction to study
instanton superpotentials on those manifolds and their quotients. We identify a
non-toric quotient of a non-favorable CICY with a single genus zero curve in a
certain homology class, so that a cancellation \`a la Beasley-Witten is not
possible. In another example, we study a non-toric quotient of a favorable CICY
and check that the superpotential still vanishes. From this and related
examples, we conjecture that the Beasley-Witten cancellation result can be
extended to toric and non-toric quotients of CICYs, but can be avoided if the
CICY is non-favorable.
