In the present work, an enhanced perturbation analysis to solve a time-fractional Klein–Gordon equation (KG equation) and obtain an analytic approximate periodic solution is examined. The Riemann–Liouville fractionalderivative is utilised. A travelling wave solution is adopted throughout the perturbation method by including two small perturbation parameters. The amplitude equation is formulated in the form of a cubic–quintic complexnonlinear Schrödinger equation. The solution of this equation leads to a transcendental frequency equation. An approximate solution to this frequency equation is performed. The stability criteria are derived. The procedure adopted here is very significant and powerful for solving many nonlinear partial differential equations (NLPDEs) arising in nonlinear science and engineering.