Am I mistaken that the first part is correct by the Burnside argument, without coprimeness? Let $V=\langle z\rangle,$ and let $W$ be a subgroup of order $p$ of $Z(P).$ Since in particular $W\subseteq P,$ there is $a\in A$ such that $V^a=W.$ Then $V$ centralizes $P$ and $P^{a^{-1}},$ so $P$ and $P^{a^{-1}}$ are Sylow $p$-subgroups of $C_G(V).$ Hence there is $b\in C_G(V)$ with $P^{a^{-1}}=P^b.$ That is, $P=P^{ba}.$ Then $ba\in N_{GA}(P)$ and $V^{ba}=V^a=W,$ as required.

I think the first part is correct as stated, by the Burnside argument. Let $V=\langle z\rangle,$ and let $W$ be a subgroup of order $p$ of $Z(P).$ Since in particular $W\subseteq P,$ there is $a\in A$ such that $V^a=W.$ Then $V$ centralizes $P$ and $P^{a^{-1}},$ so $P$ and $P^{a^{-1}}$ are Sylow $p$-subgroups of $C_G(V).$ Hence there is $b\in C_G(V)$ with $P^{a^{-1}}=P^b.$ That is, $P=P^{ba}.$ Then $ba\in N_{GA}(P)$ and $V^{ba}=V^a=W,$ as required.