Cluster Toda chains and Nekrasov functions

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.