I looked to my own old paper with Bokalo and have found there Example 9.3 answering this problems:

Example. Under Martin's Axiom (more precisely, $\mathrm{add}(\mathcal M)=\mathrm{cof}(\mathcal M)$) there exists a function $f:X\to Y$ between separable metrizable spaces, which is $G_\delta$-measurable but not $\sigma$-continuous.

I looked to my own old paper with Bokalo and have found there Example 9.3 answering this problem:

Example. Under Martin's Axiom (more precisely, $\mathrm{add}(\mathcal M)=\mathrm{cof}(\mathcal M)$) there exists a bijective function $f:X\to Y$ between zero-dimensional separable metrizable spaces such that $f^{-1}$ is continuous, $f$ is $G_\delta$-measurable but $f$ is not $\sigma$-continuous.

In fact, the problem was motivated by this question of Karlova, which is still open.