We extend previous work on symbolic self-triggered control for non-deterministic continuous-time nonlinear systems without stability assumptions to a larger class of specifications. Our goal is to synthesise a controller for two objectives: the first one is modelled as a right-recursive LTL formula, and the second one is to ensure that the communication rate between the controller and the system stays below a given threshold. We translate the control problem to solving a mean-payoff parity game played on a discrete graph. Apart from extending the class of specifications, we propose a heuristic method to shorten the computation time. Finally, we illustrate our results on the example of a navigating nonholonomic robot with several specifications.