  In this work, we study general features of a regime where gauge fields
produced during inflation cause a strong backreaction on the background
evolution and its impact on the spectrum and the correlation length of gauge
fields. With this aim, the gradient-expansion formalism previously proposed for
the description of inflationary magnetogenesis in purely kinetic or purely
axial coupling models, is extended to the case when both types of coupling are
present. As it is formulated in position space, this method allows us to
self-consistently take into account the backreaction of generated gauge fields
on the inflationary background because it captures the nonlinear evolution of
all physically relevant gauge-field modes at once. Using this extended
gradient-expansion formalism, suitable for a wide range of inflationary
magnetogenesis models, we study the gauge-field production in a specific
generalization of the Starobinsky $R^2$-model with a nonminimal coupling of
gauge fields to gravity. In the Einstein frame, this model implies, in addition
to an asymptotically flat inflaton potential, also a nontrivial form of kinetic
and axial coupling functions which decrease in time and, thus, are potentially
suitable for the generation of gauge fields with a scale-invariant or even
red-tilted power spectrum. The numerical analysis shows, however, that
backreaction, which unavoidably occurs in this model for the interesting range
of parameters, strongly alters the behavior of the spectrum and does not allow
us to obtain a sufficiently large correlation length for the magnetic field.
The oscillatory behavior of the generated field, caused by the retarded
response of the gauge field to changes of the inflaton velocity, was revealed.
