We study the impact of AGN ionised outflows on star formation in high-redshift AGN hosts, by combining NIR IFS observations, mapping the H$alpha$ emission and [OIII] outflows, with matched-resolution observations of the rest-frame FIR emission. We present high-resolution ALMA Band 7 observations of eight X-ray selected AGN at z~2 from the SUPER sample, targeting the rest-frame ~260 um continuum at ~2 kpc (0.2) resolution. We detected 6 out of 8 targets with S/N>10 in the ALMA maps, with continuum flux densities F = 0.27-2.58 mJy and FIR half-light radii Re = 0.8-2.1 kpc. The FIR Re of our sample are comparable to other AGN and star-forming galaxies at a similar redshift from the literature. However, we find that the mean FIR size in X-ray AGN (Re = 1.16+/- 0.11 kpc) is slightly smaller than in non-AGN (Re = 1.69+/-0.13 kpc). From SED fitting, we find that the main contribution to the 260 um flux density is dust heated by star formation, with < 4% contribution from AGN-heated dust and < 1% from synchrotron emission. The majority of our sample show different morphologies for the FIR (mostly due to reprocessed stellar emission) and the ionised gas emission (H$alpha$ and [OIII], mostly due to AGN emission). This could be due to the different locations of dust and ionised gas, the different sources of the emission (stars and AGN), or the effect of dust obscuration. We are unable to identify any residual H$alpha$ emission, above that dominated by AGN, that could be attributed to star formation. Under the assumption that the FIR emission is a reliable tracer of obscured star formation, we find that the obscured star formation activity in these AGN host galaxies is not clearly affected by the ionised outflows. However, we cannot rule out that star formation suppression is happening on smaller spatial scales than the ones we probe with our observations (< 2 kpc) or on different timescales.
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