Odd cycles in symmetric Cayley graphs on prime cyclic groups

Let $p$ be an odd prime and let $S\subseteq {\mathbb{Z}}_p$ be symmetric with $0\mathrm{\notin }S$. Let $\text{Cay}({\mathbb{Z}}_p,S)$ be the undirected Cayley graph on ${\mathbb{Z}}_p$ in which $x$ and $y$ are adjacent if and only if $x-y\in S$. For $1\le \mathrm{\ell }\le (p-1)\mathrm{/}2$, define $${\text{ex}}_{\text{Cay}}(C_{2\mathrm{\ell }+1},{\mathbb{Z}}_p)=\max \{\mathrm{\mid }S\mathrm{\mid }:S=-S,0\mathrm{\notin }S,\text{Cay}({\mathbb{Z}}_p,S)\text{ contains no }{C}_{2\mathrm{\ell }+1}\}\mathrm{.}$$ Confirming a conjecture of Cashman and Kelley, we prove that if $p=2\mathrm{\ell }+1$, then ${\text{ex}}_{\text{Cay}}(C_{2\mathrm{\ell }+1},{\mathbb{Z}}_p)=0$, while if $p>2\mathrm{\ell }+1$, then $${\text{ex}}_{\text{Cay}}(C_{2\mathrm{\ell }+1},{\mathbb{Z}}_p)=2\text{floor}\frac{p+2\mathrm{\ell }+1}{2(2\mathrm{\ell }+1)}\mathrm{.}$$ The proof combines a sharp additive zero-sum odd-girth argument with weak odd pancyclicity to transfer the result from odd-girth exclusion to fixed odd-cycle exclusion. We also give a canonical extremal family, an exact extremality criterion in terms of odd zero-sum avoidance, and an example showing that extremizers need not be dilates of the canonical construction.