  In this work we present updated forecasts on parameterised modifications of
gravity that can capture deviations of the behaviour of cosmological density
perturbations beyond $\Lambda$CDM. For these forecasts we adopt the SKA
Observatory (SKAO) as a benchmark for future cosmological surveys at radio
frequencies, combining a continuum survey for weak lensing and angular galaxy
clustering with an HI galaxy survey for spectroscopic galaxy clustering that
can detect baryon acoustic oscillations and redshift space distortions.
Moreover, we also add 21cm HI intensity mapping, which provides invaluable
information at higher redshifts, and can complement tomographic resolution,
thus allowing us to probe redshift-dependent deviations of modified gravity
models. For some of these cases, we combine the probes with other optical
surveys, such as the Dark Energy Spectroscopic Instrument (DESI) and the Vera
C. Rubin Observatory (VRO). We show that such synergies are powerful tools to
remove systematic effects and degeneracies in the non-linear and small-scale
modelling of the observables. Overall, we find that the combination of all SKAO
radio probes will have the ability to constrain the present value of the
functions parameterising deviations from $\Lambda$CDM ($\mu$ and $\Sigma$) with
a precision of $2.7\%$ and $1.8\%$ respectively, competitive with the
constraints expected from optical surveys and with constraints we have on
gravitational interactions in the standard model. Exploring the radio-optical
synergies, we find that the combination of VRO with SKAO can yield extremely
tight constraints on $\mu$ and $\Sigma$ ($0.9\%$ and $0.7\%$ respectively),
which are further improved when the cross-correlation between intensity mapping
and DESI galaxies is included.
