  It is quite well known for some time that string inspired axionic terms of
the form $\nu (\phi)\tilde{R}R$, known also as Chern-Simons terms, do not
affect the scalar perturbations and the background evolution for a flat
Friedman-Robertson-Walker Universe. In this paper we study and quantify the
implications of the presence of the above term in the context of vacuum $f(R)$.
Particularly, we assume that axionic dark matter is present during inflation,
and we examine in a quantitative way the effects of axionic Chern-Simons terms
on the tensor perturbations. The axion field is quantified in terms of a
canonical scalar field, with broken Peccei-Quinn symmetry. The model perfectly
describing axions as potential dark matter candidates is based on the so-called
misalignment mechanism, in which case the axion is frozen near its non-zero
vacuum expectation value during early times in which $H\gg m_a$. In effect, the
inflationary era is mainly controlled by the $f(R)$ gravity and the
Chern-Simons term. As we demonstrate, the Chern-Simons term may achieve to make
a non-viable $f(R)$ gravity theory to be phenomenologically viable, due to the
fact that the tensor-to-scalar ratio is significantly reduced, and the same
applies to the spectral index of the tensor perturbations $n_T$. Also by
studying the Starobinsky model in the presence of the Chern-Simons term, we
demonstrate that it is possible to further reduce the amount of primordial
gravitational radiation. The issues of having parity violating gravitational
waves, also the graceful exit from inflation due to axion oscillations and
finally the unification of dark energy-inflation and axion dark matter in the
same $f(R)$ gravity-axion dark matter model, are also briefly discussed.
