  Silicon nitride (Si$_3$N$_4$) is an ever-maturing integrated platform for
nonlinear optics. Yet, due to the absence of second-order ($\chi^{(2)}$)
nonlinearity, Si$_3$N$_4$ is mostly considered for third-order ($\chi^{(3)}$)
nonlinear interactions. Recently, this limitation was overcome by optical
poling in both Si$_3$N$_4$ waveguides and microresonators via the photogalvanic
effect, resulting in the inscription of quasi-phase-matched $\chi^{(2)}$
gratings. Here, we report cascaded nonlinear effects in a normal dispersion
Si$_3$N$_4$ microresonator with combined $\chi^{(2)}$ and $\chi^{(3)}$
nonlinearities. We demonstrate that the photo-induced $\chi^{(2)}$ grating also
provides phase-matching for the sum-frequency generation process, enabling the
initiation and successive switching of primary combs at pump wavelength.
Additionally, the doubly resonant pump and second-harmonic fields allow for
cascaded third-harmonic generation, where a secondary optically written
$\chi^{(2)}$ grating is identified. Finally, we reach a low-noise, broadband
microcomb state evolved from the sum-frequency coupled primary comb. These
results expand the scope of cascaded effects in $\chi^{(2)}$ and $\chi^{(3)}$
microresonators.
