We study continuous stochastic games with heterogeneous mean field interactions and jumps on large networks and explore their limit counterparts. We introduce the graphon game model based on a controlled graphon mean field stochastic differential equation system with jumps, which can be regarded as the limiting case of a finite game dynamic system as the number of players goes to infinity. We examine the case of controlled dynamics, with control terms present in the drift, diffusion, and jump components. We focus on the study of Markovian controls and concentrate on the limit theory. We provide convergence results on the state trajectories and their laws, transitioning from finite game systems to graphon systems. We also study approximate equilibria for finite games on large networks, using the graphon equilibrium as a benchmark. The rates of convergence are analyzed under various underlying graphon models and regularity assumptions.