We introduce a second-order gyro-averaged extension to a guiding-center transport model for test particles subject to an $\mathbf{E}\times\mathbf{B}$ drift in a tokamak. Using this approximation, we numerically investigate the influence of finite Larmor radius effects on the chaotic transport and the formation of transport barriers. For non-monotonic plasma profiles, we show that the twist condition of the system is violated along a curve, usually called shearless transport barrier, which, under non-equilibrium conditions, exhibits significant resilience to destruction, thereby inhibiting chaotic transport. We analyze bifurcation diagrams of the shearless barriers and escape rates to explore the impact of the Larmor radius on controlling chaotic transport while varying the amplitude of electrostatic perturbations. Our findings show that increasing the Larmor radius enhances the robustness of transport barriers, as a larger electrostatic perturbation amplitude is required to disrupt them. Additionally, even in the absence of transport barriers, we observe a decrease in chaotic transport with an increase in the Larmor radius.