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

For $i\geq2$, we compute the first coefficients of the class $[\overline{D}(\mu;3)]$ in the rational Picard group of the moduli of Prym curves $\overline{\mathcal{R}}_{2i}$, where $D(\mu;3)$ is the divisor parametrizing pairs $[C,\eta]$ for which there exists a degree $2i$ map $\pi\colon C\rightarrow \mathbb{P}^1$ having ramification profile $(2,\ldots,2)$ 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.