We propose an extended Su-Schrieffer-Heeger chain that integrates long-range hopping and a synthetic Peierls phase, providing a tunable circuit platform to trigger and control non-Hermitian transitions and topological phase transitions. We show that the long-range hopping strength acts as a trigger for topological phase transitions, destabilizing edge states through band-gap closure. Simultaneously, the Peierls phase serves as a synthetic gauge field, enabling active and reversible switching between trivial and topological regimes via phase-selective coupling. Under balanced gain and loss, topological edge states remain robust but undergo dramatic reconstruction at exceptional points, accompanied by non-Hermitian skin effects and complex spectral loops. Furthermore, we discover rich wave-packet dynamics under parameter tuning, including phase-dependent localization, spectral winding, and anomalous diffusion near exceptional points. Our work establishes a highly tunable circuit framework for dynamically coupling topological, non-Hermitian, and transport phenomena, offering direct pathways toward robust topological photonics, enhanced sensing, and synthetic quantum matter.

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• Circuit quantum electrodynamics
• Long-range interactions
• Quantum optics
• Topological phase transition
• Non-Hermitian systems