  We study galaxies in JADES Deep to study the evolution of the ionising photon
production efficiency, $\xi_{\rm{ion}}$, observed to increase with redshift. We
estimate $\xi_{\rm{ion}}$ for a sample of 677 galaxies at $z \sim 4 - 9$ using
NIRCam photometry. Specifically, combinations of the medium and wide bands
F335M-F356W and F410M-F444W to constrain emission lines that trace
$\xi_{\rm{ion}}$: H$\alpha$ and [OIII]. Additionally, we use the spectral
energy distribution fitting code \texttt{Prospector} to fit all available
photometry and infer galaxy properties. The flux measurements obtained via
photometry are consistent with FRESCO and NIRSpec-derived fluxes. Moreover, the
emission-line-inferred measurements are in tight agreement with the
\texttt{Prospector} estimates. We also confirm the observed $\xi_{\rm{ion}}$
trend with redshift and M$_{\rm{UV}}$, and find: $\log \xi_{\rm{ion}}
(z,\text{M}_{\rm{UV}}) = (0.05 \pm 0.02)z + (0.11 \pm 0.02) \text{M}_{\rm{UV}}
+ (27.33 \pm 0.37)$. We use \texttt{Prospector} to investigate correlations of
$\xi_{\rm{ion}}$ with other galaxy properties. We see a clear correlation
between $\xi_{\rm{ion}}$ and burstiness in the star formation history of
galaxies, given by the ratio of recent to older star formation, where
burstiness is more prevalent at lower stellar masses. We also convolve our
$\xi_{\rm{ion}}$ relations with luminosity functions from the literature, and
constant escape fractions of 10 and 20\%, to place constraints on the cosmic
ionising photon budget. By combining our results, we find that if our sample is
representative of the faint low-mass galaxy population, galaxies with bursty
star formation are efficient enough in producing ionising photons and could be
responsible for the reionisation of the Universe.
