Context. Migration of giant planets in discs with low viscosity has been studied recently. Results have shown that the proportionality between migration speed and the disc’s viscosity is broken by the presence of vortices that appear at the edges of the planet-induced gap. Under some conditions, this ‘vortex-driven’ migration can be very slow and eventually stops. However, this result has been obtained for discs whose radial mass transport is too low (due to the small viscosity) to be compatible with the mass accretion rates that are typically observed for young stars. Aims. Our goal is to investigate vortex-driven migration in low-viscosity discs in the presence of radial advection of gas, as expected from angular momentum removal by magnetised disc winds. Methods. We performed three dimensional simulations using the grid-based code FARGOCA. We mimicked the effects of a disc wind by applying a synthetic torque on a surface layer of the disc characterised by a prescribed column density Σ_A so that it results in a disc accretion rate of Ṁ_A = 10⁻⁸ M_⊙ yr⁻¹. We have considered values of Σ_A typical of the penetration depths of different ionising processes. Discs with this structure are called ‘layered’ and the layer where the torque is applied is denoted as ‘active’. We also consider the case of accretion focussed near the disc midplane to mimic transport properties induced by a large Hall effect or by weak Ohmic diffusion. Results. We observe two migration phases: in the first phase, which is exhibited by all simulations, the migration of the planet is driven by the vortex and is directed inwards. This phase ends when the vortex disappears after having opened a secondary gap, as is typically observed in vortex-driven migration. Migration in the second phase depends on the ability of the torque from the planet to block the accretion flow. When the flow is fast and unimpeded, corresponding to small Σ_A, migration is very slow, similar to when there is no accreting layer in the disc. When the accretion flow is completely blocked, migration is faster (typically ṙ_p ~ 12 AU Myr⁻¹ at 5 au) and the speed is controlled by the rate at which the accretion flow refills the gap behind the migrating planet. The transition between the two regimes, occurs at Σ_A ~ 0.2 g cm⁻² and 0.65 g cm⁻² for Jupiter or Saturn mass planets at 5.2 au, respectively. Conclusions. The migration speed of a giant planet in a layered protoplanetary disc depends on the thickness of the accreting layer. The lack of large-scale migration apparently experienced by the majority of giant exoplanets can be explained if the accreting layer is sufficiently thin to allow unimpeded accretion through the disc.