  We continue our study of one-loop integrals associated to BPS-saturated
amplitudes in $\mathcal{N}=2$ heterotic vacua. We compute their large-volume
behaviour, and express them as Fourier series in the complexified volume, with
Fourier coefficients given in terms of Niebur-Poincar\'e series in the complex
structure modulus. The closure of Niebur-Poincar\'e series under modular
derivatives implies that such integrals derive from holomorphic prepotentials
$f_n$, generalising the familiar prepotential of $\mathcal{N}=2$ supergravity.
These holomorphic prepotentials transform anomalously under T-duality, in a way
characteristic of Eichler integrals. We use this observation to compute their
quantum monodromies under the duality group. We extend the analysis to modular
integrals with respect to Hecke congruence subgroups, which naturally arise in
compactifications on non-factorisable tori and freely-acting orbifolds. In this
case, we derive new explicit results including closed-form expressions for
integrals involving the ${\varGamma}_0(N)$ Hauptmodul, a full characterisation
of holomorphic prepotentials including their quantum monodromies, as well as
concrete formulae for holomorphic Yukawa couplings.
