  Superbursts from accreting neutron stars probe nuclear reactions at extreme
densities ($\rho \approx 10^{9}~g\,cm^{-3}$) and temperatures ($T>10^9~K$).
These bursts ($\sim$1000 times more energetic than type I X-ray bursts) are
most likely triggered by unstable ignition of carbon in a sea of heavy nuclei
made during the rp-process of regular type I X-ray bursts (where the
accumulated hydrogen and helium are burned). An open question is the origin of
sufficient amounts of carbon, which is largely destroyed during the rp-process
in X-ray bursts. We explore carbon production in steady-state burning via the
rp-process, which might occur together with unstable burning in systems showing
superbursts. We find that for a wide range of accretion rates and accreted
helium mass fractions large amounts of carbon are produced, even for systems
that accrete solar composition. This makes stable hydrogen and helium burning a
viable source of carbon to trigger superbursts. We also investigate the
sensitivity of the results to nuclear reactions. We find that the
$^{14}$O($\alpha$,p)$^{17}$F reaction rate introduces by far the largest
uncertainties in the $^{12}$C yield.
