  We compute odd-degree genus 1 quasimap (and Gromov--Witten) invariants of
moduli spaces of Higgs $\mathrm{SL}_2$-bundles on a curve of genus $g\geq2$. We
also compute certain invariants for all prime ranks. This proves some parts of
author's conjectures on quasimap invariants of moduli spaces of Higgs bundles.
More generally, our methods provide a computation scheme for genus 1 quasimap
(and Gromov--Witten) invariants in the case when degrees of maps are coprime to
the rank. This requires a careful analysis of the localisation formula for
certain Quot schemes parametrising higher-rank quotients on an elliptic curve.
Invariants for degrees which are not coprime to the rank exhibit a very
different structure for a reason that we explain.
