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 ($M_{rm Ni} = 0.4pm0.2$~M$_{odot}$, $M_{rm ej} = 5pm2$~M$_{odot}$, and $E_{rm K} = 2pm1 times 10^{52}$ 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 $E_{rm K}$--$Gammabeta$~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 $bar{E}_{rm K} = 2.1times10^{52}$ 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.
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