Fields of moduli and fields of definition of odd signature curves

Let $X$ be a smooth projective algebraic curve of genus $g\ge 2$ defined over a field $K$. We show that $X$ can be defined over its field of moduli if it has odd signature, i.e. if the signature of the covering $X\to X\mathrm{/}\text{Aut}(X)$ is of type $(0;c_1,\mathrm{.}\mathrm{.}\mathrm{.},c_k)$, where some $c_i$ appears an odd number of times. This result is applied to $q$-gonal curves and to plane quartics. For $q$-gonal curves, we prove that non-normal $q$-gonal curves can be defined over their field of moduli and we construct examples of normal $q$-gonal curves with field of moduli $\mathbb{R}$ that can not be defined over $\mathbb{R}$. For plane quartics, we prove that they can be defined over their field of moduli if the automorphism group is not isomorphic to either $C_2$ or $C_2\times C_2$.