quantum field theory - Some questions about Ward-Takahashi Identity

I'm a learner of Peskin and Schroeder's textbook of quantum field theory.

I have proceeded to Ward-Takahashi identity and have one question when I look for Wikipedia for reference.

The following is the Ward-Takahashi identity, where $M^{\mu}$ is the correlation function for $n$ inserting electrons and $n$ out-going electrons.

$$k_{\mu}M^{\mu}(k;p_1...p_n;q_1...q_n)=-e\sum_i[M_0(k;p_1...p_n;q_1...(q_i-k)...q_n)-M_0(k;p_1...(p_i+k)...p_n;q_1...q_n)].$$

The wiki says that

Note that if ($M^{\mu}$) has its external electrons on-shell, then the amplitudes on the right-hand side of this identity each have one external particle off-shell, and therefore they do not contribute to S-matrix elements.

Does on-shell means a divergent contribution according to LSZ reduction formula?

Besides, can you tell me why the S matrix is zero if all the external electrons in the left hand side are on shell?