  We present optical and near-infrared photometry of GRB~140606B ($z=0.384$),
and optical photometry and spectroscopy of its associated supernova (SN). The
results of our modelling indicate that the bolometric properties of the SN
($M_{\rm Ni} = 0.4\pm0.2$~M$_{\odot}$, $M_{\rm ej} = 5\pm2$~M$_{\odot}$, and
$E_{\rm K} = 2\pm1 \times 10^{52}$ erg) are fully consistent with the
statistical averages determined for other GRB-SNe. However, in terms of its
$\gamma$-ray emission, GRB~140606B is an outlier of the Amati relation, and
occupies the same region as low-luminosity ($ll$) and short GRBs. The
$\gamma$-ray emission in $ll$GRBs is thought to arise in some or all events
from a shock-breakout (SBO), rather than from a jet. The measured peak photon
energy ($E_{\rm p}\approx800$ keV) is close to that expected for $\gamma$-rays
created by a SBO ($\gtrsim1$ MeV). Moreover, based on its position in the
$M_{V,\rm p}$--$L_{\rm iso,\gamma}$~plane and the $E_{\rm
K}$--$\Gamma\beta$~plane, GRB~140606B has properties similar to both SBO-GRBs
and jetted-GRBs. Additionally, we searched for correlations between the
isotropic $\gamma$-ray emission and the bolometric properties of a sample of
GRB-SNe, finding that no statistically significant correlation is present. The
average kinetic energy of the sample is $\bar{E}_{\rm K} = 2.1\times10^{52}$
erg. All of the GRB-SNe in our sample, with the exception of SN 2006aj, are
within this range, which has implications for the total energy budget available
to power both the relativistic and non-relativistic components in a GRB-SN
event.
