  Nowadays, efforts are being devoted to the study of alternative cosmological
scenarios, in which, modifications of General Relativity (GR) theory have been
proposed to explain the late cosmic acceleration, without assuming the
existence of the dark energy (DE) component. We investigate the $R^2$-corrected
Appleby-Battye model, or $R^2$-AB model, which consists of an $f(R)$ model with
only one extra free parameter $b$, besides the cosmological parameters of the
flat-$\Lambda$CDM model: $H_0$ and $\Omega_{m,0}$. Regarding this model, it was
already shown that a positive value for $b$ is required for the model to be
consistent with Solar System tests, moreover, the condition for the existence
of a de~Sitter state requires $b \ge 1.6$. To impose observational constraints
on the $R^2$-AB model, we consider in our analyses two data sets: cosmic
chronometer $H(z)$ data for the background level, and $[f\sigma_8](z)$ data,
for the perturbative level. The first one provides $b = 1.6^{+3.1}_{-0.0}$ and
the cosmological parameters $\{H_0 ,\Omega_{m,0}\}$ in agreement to Planck
values, while the second one, indicates $b = 1.76^{+2.91}_{-0.15}$ and the
parameters $\{\Omega_{m,0},\sigma_{8,0} \}$ also in agreement to Planck values;
in the last case the data was marginalized over the parameter $H_0$.
Additionally, we perform illustrative analyses that compare this $f(R)$ model
with the flat-$\Lambda$CDM model, considering several values of the parameter
$b$, for diverse cosmological functions like the Hubble function $H(z)$, the
equation of state $w_{eff}(z)$, the parametrized growth rate of cosmic
structures $[f \sigma_8](z)$, and $\sigma_8(z)$.
