  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.
