  We explore the relationship between the effective dynamics in standard loop
quantum cosmology (LQC) based on holonomies and triads obtained from
gauge-fixing fluxes, and a modification of LQC based on holonomies and
gauge-covariant fluxes (referred to as gLQC). Both the models yield singularity
resolution via a bounce because of non-perturbative quantum geometric effects
resulting in a maximum for energy density. In LQC, the bounce is extremely well
captured by a $\rho^2$ term in energy density with a negative sign which
emerges as a non-perturbative modification to the classical Friedmann and
Raychaudhuri equations. But, details of such modifications in gLQC have
remained hidden due to an arduous nature of gauge-covariant flux modifications
which do not allow writing above equations in a closed form. To extract these
modifications we explore the large volume, weak curvature limit for matter with
a fixed equation of state and obtain higher order corrections to the classical
theory. We find that in the weak curvature limit of gLQC, in the post-bounce
branch, the first order correction beyond classical theory fully recovers the
form of modified Friedmann and Raychaudhuri equations of LQC. In contrast, due
to an asymmetric bounce in gLQC, the weak curvature limit of the pre-bounce
branch exhibits a novel structure with a $\rho^{3/2}$ term as a first order
correction beyond classical theory while the $\rho^2$ term appears as a second
order correction. Our work shows that gLQC has a far richer structure which
includes the form of dynamical equations with non-perturbative modifications in
LQC in its weak curvature limit. This indicates that more general loop
quantizations of cosmological sectors can reveal LQC at some truncation, and
possibly there exist a tower of potentially interesting higher order
modifications from quantum geometry which are hidden in the setting of LQC.
