Prym enumerative geometry and a Hurwitz divisor in ${\bar{\mathcal{R}}}_{2i}$

For $i\ge 2$, we compute the first coefficients of the class $[\bar{D}(\mu ;3)]$ in the rational Picard group of the moduli of Prym curves ${\bar{\mathcal{R}}}_{2i}$, where $D(\mu ;3)$ is the divisor parametrizing pairs $[C,\eta ]$ for which there exists a degree $2i$ map $\pi \text{\hspace{0.17em}⁣}:C\to {\mathbb{P}}^1$ having ramification profile $(2,\dots ,2)$ above two points $q_1,q_2$, a triple ramification somewhere else and satisfying $$\mathcal{O}$_C(\frac{\pi^{}(q_1)-\pi^{}(q_2)}{2})\cong \eta$. Furthermore, we provide several new Prym enumerative results related to this situation.