Finding an MST with one adding and removing vertex operation
After long hours facing this problem, with my advisor help, we could solve this problem in $O(|E(G^{'})| log\text{ }C)$ time complexity by using one iteration of the Boruvka's algorithm, where $C$ is the number of connected components of $G^{''} (V(G^{'}), T\backslash \{(v_k, v_i): v_k \in V(G^{'})\})$, i.e., the number of connected components of the MST $T$ by removing $v_i$ (and edges from $v_i$ to every other vertex in $T$) plus one (due to the inserted vertex $v_j$).
Note that, this solution might have $O(|E(G^{'})| log\text{ }|V(G^{'})|)$ time complexity if the removing of $v_i$ in $T$ (and edges from $v_i$ to every other vertex in $T$) lead to a totally disconnected graph. Also, note that, this solution has $\Omega(|E(G^{'})|)$ if C is equals to 3.