  Assuming that a scalar field controls the inflationary era, we examine the
combined effects of string and $f(R)$ gravity corrections on the inflationary
dynamics of canonical scalar field inflation, imposing the constraint that the
speed of the primordial gravitational waves is equal to that of light's.
Particularly, we study the inflationary dynamics of an Einstein-Gauss-Bonnet
gravity in the presence of $\alpha R^2$ corrections, where $\alpha$ is a free
coupling parameter. As it was the case in the pure Einstein-Gauss-Bonnet
gravity, the realization that the gravitational waves propagate through
spacetime with the velocity of light, imposes the constraint that the
Gauss-Bonnet coupling function $\xi(\phi)$ obeys the differential equation
$\ddot\xi=H\dot\xi$, where $H$ is the Hubble rate. Subsequently, a relation for
the time derivative of the scalar field is extracted which implies that the
scalar functions of the model, which are the Gauss-Bonnet coupling and the
scalar potential, are interconnected and simply designating one of them
specifies the other immediately. In this framework, it is useful to freely
designate $\xi(\phi)$ and extract the corresponding scalar potential from the
equations of motion but the opposite is still feasible. We demonstrate that the
model can produce a viable inflationary phenomenology and for a wide range of
the free parameters. Also, a mentionable issue is that when the coupling
parameter $\alpha$ of the $R^2$ correction term is $\alpha<10^{-3}$ in Planck
Units, the $R^2$ term is practically negligible and one obtains the same
equations of motion as in the pure Einstein-Gauss-Bonnet theory, however the
dynamics still change, since now the time derivative of $\frac{\partial
f}{\partial R}$ is nonzero.
