Continuity of measurable cocycles

Suppose $G\curvearrowright X$ is a Polish group action, $H$ is a Polish group and $G\times X\overset{\psi}\longrightarrow H$ is a cocycle that is continuous in the second variable. If $\psi$ is either Baire measurable or is $\lambda\times \mu$-measurable with respect to a Haar measure $\lambda$ on $G$ and a fully supported $\sigma$-finite Borel measure $\mu$ on $X$, then $\psi$ is jointly continuous.