Borel sets with countable sections for nonseparable spaces

We prove that every (extended) Borel subset $E$ of $X\times Y$, where $X$ is a complete metric and $Y$ is Polish, can be covered by countably many extended Borel graphs of mappings from $X$ to $Y$ if the sections $E_x=\{y\in Y:(x,y)\in E\}$, $x\in X$, are countable. This is a nonseparable version of a classical theorem of Luzin and Novikov.