Regular dessins with moduli fields of the form $\mathbb{Q}(\zeta_p,\sqrt[p]{q})$

Gareth Jones asked during the 2014 SIGMAP conference for examples of regular dessins with nonabelian fields of moduli. In this paper, we first construct dessins whose moduli fields are nonabelian Galois extensions of the form $\mathbb{Q}(\zeta_p,\sqrt[p]{q})$, where $p$ is an odd prime and $\zeta_p$ is a $p$th root of unity and $q\in\mathbb{Q}$ is not a $p$th power, and we then show that their regular closures have the same moduli fields. Finally, in the special case $p=q=3$ we give another example of a regular dessin with moduli field $\mathbb{Q}(\zeta_3,\sqrt[3]{3})$ of degree $2^{19}\cdot3^4$ and genus $14155777$.