  The motto of this work is to generate a general formalism of $f(\bar{R},
L(X))-$gravity in the context of dark energy under the framework of the {\bf
K-}essence emergent geometry with the Dirac-Born-Infeld (DBI) variety of
action, where $\bar{R}$ is the familiar Ricci scalar, $L(X)$ is the DBI type
non-canonical Lagrangian with $X={1\over
2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$ and $\phi$ is the {\bf K-}essence
scalar field. The emergent gravity metric $\bar{G}_{\mu\nu}$ and the well known
gravitational metric $g_{\mu\nu}$ are not conformally equivalent. We have
constructed a modified field equation using the metric formalism in $f(\bar{R},
L(X))$-gravity incorporating the corresponding Friedmann equations in the
framework of the background gravitational metric which is of
Friedmann-Lema{\^i}tre-Robertson-Walker (FLRW) type. The solution of modified
Friedmann equations have been deduced for the specific choice of $f(\bar{R},
L(X))$, which is of Starobinsky-type, using power law expansion method. The
consistency of the model with the accelerating phase of the Universe has been
shown, when we restrict ourselves to consider the value of the dark energy
density, as $\dot\phi^{2}=\frac{8}{9}=0.888 <1$, which indicates that the
present Universe is dark energy dominated. Graphical plots for the energy
density ($\rho$), pressure ($p$) and equation of state parameter ($\omega$)
w.r.t. time ($t$) based on parametric values are interestingly consistent with
the dark energy domination and hence accelerating features. We also put some
light on the corresponding energy conditions and constraints of the $f(\bar{R},
L(X))$ theory with one basic example.
