If you know $\pi$ then $$\begin{align*} m_{k}=E_{k}[T_{k}^{(1)}]=\frac{1}{\pi_{k}}\\\\ \gamma_{i}^{k}=\frac{\pi_{i}}{\pi_{k}} \end{align*}$$ In a sense, $\gamma_{i}^{k}$​ reflects the proportion of time the chain spends in state $i$ relative to state $k$ in the long run, which is a key concept in determining the invariant distribution.