We study the cyclic dominance of three species in two-dimensional constrained Newman-Watts networks with a four-state variant of the rock-paper-scissors game. By limiting the maximal connection distance Rmax in Newman-Watts networks with the long-range connection probability p , we depict more realistically the stochastic interactions among species within ecosystems. When we fix mobility and vary the value of p or Rmax, the Monte Carlo simulations show that the spiral waves grow in size, and the system becomes unstable and biodiversity is lost with increasing p or Rmax. These results are similar to recent results of Reichenbach et al. [Nature (London) 448, 1046 (2007)], in which they increase the mobility only without including long-range interactions. We compared extinctions with or without long-range connections and computed spatial correlation functions and correlation length. We conclude that long-range connections could improve the mobility of species, drastically changing their crossover to extinction and making the system more unstable.