Does there exist a quasisimple group $G$ and an odd prime $p$ such that $G$ has cyclic Sylow $p$-subgroups and a weakly real element of $p$-power order?

From Strongly real elements of odd order in sporadic finite simple groups the only sporadic finite simple group which has weakly real $2$-regular elements is McL. But McL does not have cyclic Sylow $p$-subgroups for the prime orders $3$ and $5$ of the weakly real elements.