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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 $X={1\over 2}g^{μν}\nabla_μϕ\nabla_νϕ$ and $ϕ$ is the {\bf K-}essence scalar field. The emergent gravity metric $\bar{G}_{μν}$ and the well known gravitational metric $g_{μν}$ 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{î}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 $\dotϕ^{2}=\frac{8}{9}=0.888 <1$, which indicates that the present Universe is dark energy dominated. Graphical plots for the energy density ($ρ$), pressure ($p$) and equation of state parameter ($ω$) 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.