We know that $PV = n RT$ and are given $P = kV^2$.

$\Rightarrow PV = nRT = kV^3$

Differentiating, we get $nR dT = 3kV^2dV \rightarrow 3PdV = nRdT$

Now, $dQ = dU + pDV$

$\Rightarrow nCdT = nC_vdT +\large\frac{nRdT}{3}$

$\Rightarrow C = C_v + R/3$

Note: We can also substitute $C_v = 3R/2$ such that $ C = 11R/6$