  In this paper the relation between the cluster integrable systems and
$q$-difference equations is extended beyond the Painlev\'e case.
  We consider the class of hyperelliptic curves when the Newton polygons
contain only four boundary points. The corresponding cluster integrable Toda
systems are presented, and their discrete automorphisms are identified with
certain reductions of the Hirota difference equation. We also construct
non-autonomous versions of these equations and find that their solutions are
expressed in terms of 5d Nekrasov functions with the Chern-Simons
contributions, while in the autonomous case these equations are solved in terms
of the Riemann theta-functions.
