  We study the multi-instanton partition functions of the $\Omega$-deformed
$\mathcal N =2^{*}$ $SU(2) $ gauge theory in the Nekrasov-Shatashvili (NS)
limit. They depend on the deformation parameters $\epsilon_{1}$, the scalar
field expectation value $a$, and the hypermultiplet mass $m$. At fixed
instanton number $k$, they are rational functions of $\epsilon_{1}, a, m$ and
we look for possible regularities that admit a parametrical description in the
number of instantons. In each instanton sector, the contribution to the
deformed Nekrasov prepotential has poles for "large" deformation parameters. To
clarify the properties of these singularities we exploit Bethe/gauge
correspondence and examine the special ratios $m/\epsilon_{1}$ at which the
associated spectral problem is $n$-gap. At these special points we illustrate
several structural simplifications occurring in the partition functions. After
discussing various tools to compute the prepotential, we analyze the
non-perturbative corrections up to $k=24$ instantons and present various closed
expressions for the coefficients of the singular terms. Both the regular and
singular parts of the prepotential are resummed over all instantons and
compared successfully with the exact prediction from the spectral theory of the
Lam\'e equation, showing that the pole singularities are an artifact of the
instanton expansion. The analysis is fully worked out in the 1-gap case, but
the final pole cancellation is proved for a generic ratio $m/\epsilon_{1}$
relating it to the gap width of the Lam\'e equation.
