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In this paper, we focus on studying the long time behaviors of a type ofrandom walk called the $\delta$ once-reinforced random walk ($\delta$-ORRW) ona finite connected graph $G$ with at least 3 vertices for $\delta>0$. Thisrandom walk process involves edge weights and the traversal of edges over time,where the edge weight function at the $(k+1)$-th step is given by  \[  w_k(e)=1+(\delta-1)\cdot I_{\{N(e,k)>0 \}} {\rm\ equals\ }  1 \ {\rm if}\ N(e,k)=0,\ {\rm and\ equals\ }  \delta \ {\rm if}\ N(e,k)>0,  \] and $N(e,k)$ is the number of the traversal of edge $e$ before time $k$.Our main goal is to establish a large deviation principle (LDP) for theempirical measures of this random walk and to determine the critical exponentfor the exponential integrability of certain stopping times, including thecover time.  To prove the LDP, we introduce a set of functionals related to the empiricalmeasure processes and solve a new dynamic programming equation associated withthese functionals using a variational representation. This enables us to derivea variational formula for the limit of these functionals, which is crucial forcompleting the proof of the LDP. Additionally, we demonstrate that the ratefunction $I_\delta$ of the LDP is continuous and decreasing in$\delta\in[1,\infty)$, remains constant in $\delta\in(0,1]$, and is notdifferentiable at $\delta=1$. Furthermore, we show that$I_{\delta_1}\not=I_{\delta_2}$ for any $1\leq\delta_1<\delta_2$.  Regarding the critical exponent, we establish its continuity and strictdecrease in $\delta\in(0,\infty)$, and we determine that there is noexponential integrability at this critical exponent. We also find that thelimit of the critical exponent as $\delta\to0$ is infinite if and only if $G$is a 3-vertex connected graph or a star-shaped graph.

Large deviation principle for empirical measures of once-reinforced random walks on finite graphs