  Upcoming precision measurements of the temperature anisotropy of the cosmic
microwave background (CMB) at high multipoles will need to be complemented by a
more complete understanding of recombination, which determines the damping of
anisotropies on these scales. This is the third in a series of papers
describing an accurate theory of HeI and HeII recombination. Here we describe
the effect of Thomson scattering, the $^3$He isotope shift, the contribution of
rare decays, collisional processes, and peculiar motion. These effects are
found to be negligible: Thomson and $^3$He scattering modify the free electron
fraction $x_e$ at the level of several $\times 10^{-4}$. The uncertainty in the
$2^3P^o-1^1S$ rate is significant, and for conservative estimates gives
uncertainties in $x_e$ of order $10^{-3}$. We describe several convergence
tests for the atomic level code and its inputs, derive an overall $C_\ell$
error budget, and relate shifts in $x_e(z)$ to the changes in $C_\ell$, which
are at the level of 0.5% at $\ell =3000$. Finally, we summarize the main
corrections developed thus far. The remaining uncertainty from known effects is
$\sim 0.3%$ in $x_e$.
