Objective: This paper reviews cubic–quartic Fokas–Lenells equation with perturbation terms having power-law of nonlinear refractive index for the first time. Methods: Six integration algorithms granted success with this retrieval. Results: Mathematical formulations yield bright and singular solitons with the model along with their corresponding existence criterion. Conclusion: The novelty of this paper is the extension of Kerr law to power-law and that too for cubic–quartic solitons that have emerged from the model. It has been verified if the power law nonlinearity parameter collapses to unity, all results of the paper reduce to the ones for Kerr law which are visible all over.