  In this work, we have reconstructed the extended $f(\mathcal{P})$ cubic
gravity and symmetric $f(\mathcal{Q})$ teleparallel gravity from $(m,n)$-type
Barrow Holographic Dark Energy (BHDE) and find the unknown functions
$f(\mathcal{P})$ and $f(\mathcal{Q})$ in terms of $\mathcal{P}$ and
$\mathcal{Q}$ by taking the universe to be flat, homogeneous and isotropic. We
then analyzed the behavior and stability of each model for the entire stages of
the evolution of the universe by studying several important parameters such as
the deceleration parameter, equation of state (EoS) parameter $\omega_{DE}$,
square of the speed of sound $v_s^2$. Apart from this, we have studied the
cosmographic behavior by plotting the jerk parameter, snap parameter, and lerk
parameter against the redshift. We have also examined the
$\omega'_{DE}-\omega_{DE}$ phase plane and $(r,s^*)$, $(r,q)$ statefinder
parameters that provide valuable insights into the dynamics of the universe and
the distinctive features of the dark energy. All these analyses pointed out
that our model can produce a universe going through an accelerated expansion
with the quintessence type dark energy.
