  In recent years, the idea of sub- and super-Chandrasekhar limiting mass white
dwarfs (WDs), which are potential candidates to produce under- and
over-luminous type Ia supernovae, respectively, has been a key interest in the
scientific community. Although researchers have proposed different models to
explain these peculiar objects, modified theories of Einstein's gravity,
particularly $f(R)$ gravity with $R$ being the scalar curvature, seems to be
one of the finest choices to explain both the regimes of these peculiar WDs. It
was already shown that considering higher-order corrections to the Starobinsky
model with two parameters, the structure of sub- and super-Chandrasekhar
progenitor WDs can be explained self consistently. It is also well-known that
WDs can be considered Newtonian objects because of their large size. In this
paper, we derive the weak-field limit of $f(R)$ gravity, which turns out to be
the higher-order correction to the Poisson equation. Later, we use this
equation to obtain the structures of sub- and super-Chandrasekhar limiting mass
WDs at various central densities incorporating just one model parameter.
