Given a network flow find if there's a min cut that only one of the given edges lay on it
First, compute the minimum cut value $c$ of the network.

Second, remove edge $e_1$ and increase the capacity of $e_2$ to infinity, then compute the minimum cut value $c_1$ of the new network.

Third, remove edge $e_2$ and increase the capacity of $e_1$ to infinity, then compute the minimum cut value $c_2$ of the new network.

Now you can check if $c-c_1\ge c(e_1)$ or $c-c_2\ge c(e_2)$ to see if there exists a min-cut in the initial network that contains exactly one of $e_1,e_2$.