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The dynamics of nonlinear flip-flop quantum walk with amplitude-dependent phase shifts with pertubing potential barrier is investigated. Through the adjustment between uniform local perturbations and a Kerrlike nonlinearity of the medium we find a rich set of dynamic profiles. We will show the existence of different Hadamard quantum walking regimes, including those with mobile soliton-like structures or self-trapped states. The latter is predominant for perturbations with amplitudes that tend to $φ\rightarrow π/2$. In this system, the qubit shows an unusual behavior as we increase the amplitudes of the potential barriers, and displays a monotonic decrease in the self-trapping $φ_c$ with respect to the nonlinear parameter. A chaotic-like regime becomes predominant for intermediate nonlinearity values. Furthermore, we show that by changing the quantum coins ($θ$) a non-trivial dynamic arises, where coins close to Pauli-X drives the system to a regime with predominant soliton-like structures, while the chaotic behavior are restricted to a narrow region in the $χ$-$φ$ plane. We believe that is possible to implement and observe the proprieties of this model in a integrated photonic system.