Context. High-contrast imaging (HCI) observations of exoplanets can be limited by the island effect (IE). The IE occurs when the main wavefront sensor (WFS) cannot measure sharp phase discontinuities across the telescope's secondary mirror support structures (also known as spiders). On the current generation of telescopes, the IE becomes a severe problem when the ground wind speed is below a few meters per second. During these conditions, the air that is in close contact with the spiders cools down and is not blown away. This can create a sharp optical path length difference between light passing on opposite sides of the spiders. Such an IE aberration is not measured by the WFS and is therefore left uncorrected. This is referred to as the low-wind effect (LWE). The LWE severely distorts the point spread function (PSF), significantly lowering the Strehl ratio and degrading the contrast. Aims. In this article, we aim to show that the focal-plane wavefront sensing (FPWFS) algorithm, Fast and Furious (F&F), can be used to measure and correct the IE/LWE. The F&F algorithm is a sequential phase diversity algorithm and a software-only solution to FPWFS that only requires access to images of non-coronagraphic PSFs and control of the deformable mirror. Methods. We deployed the algorithm on the SCExAO HCI instrument at the Subaru Telescope using the internal near-infrared camera in H-band. We tested with the internal source to verify that F&F can correct a wide variety of LWE phase screens. Subsequently, F&F was deployed on-sky to test its performance with the full end-to-end system and atmospheric turbulence. The performance of the algorithm was evaluated by two metrics based on the PSF quality: (1) the Strehl ratio approximation (SRA), and (2) variance of the normalized first Airy ring (VAR). The VAR measures the distortion of the first Airy ring, and is used to quantify PSF improvements that do not or barely affect the PSF core (e.g., during challenging atmospheric conditions). Results. The internal source results show that F&F can correct a wide range of LWE phase screens. Random LWE phase screens with a peak-to-valley wavefront error between 0.4µm and 2µm were all corrected to a SRA>90% and an VAR≤0.05. Furthermore, the on-sky results show that F&F is able to improve the PSF quality during very challenging atmospheric conditions (1.3-1.4''seeing at 500 nm). Closed-loop tests show that F&F is able to improve the VAR from 0.27-0.03 and therefore significantly improve the symmetry of the PSF. Simultaneous observations of the PSF in the optical (λ=750nm, Δλ=50 nm) show that during these tests we were correcting aberrations common to the optical and NIR paths within SCExAO. We could not conclusively determine if we were correcting the LWE and/or (quasi-)static aberrations upstream of SCExAO. Conclusions. The F&F algorithm is a promising focal-plane wavefront sensing technique that has now been successfully tested on-sky. Going forward, the algorithm is suitable for incorporation into observing modes, which will enable PSFs of higher quality and stability during science observations.