Energy of $\gamma$ ray $\left[E_{\mathrm{\gamma }}\right]=hv$

Momentum of $\gamma$ ray $\left[P_{\mathrm{\gamma }}\right]=\frac{h}{\lambda }=\frac{hv}{c}$
Total momentum is conserved.

${\overrightarrow{P}}_{\mathrm{\gamma }}+{\overrightarrow{P}}_{\mathrm{Nu}}=0$
where ${\overrightarrow{P}}_{\mathrm{Nu}}=$ momentum of decayed nuclei

$\Rightarrow P_{\mathrm{\gamma }}=P_{\mathrm{Nu}}$
$\Rightarrow \frac{hv}{c}=P_{\mathrm{Nu}}$
$\Rightarrow \mathrm{KE}$ of nuclei

$=\frac{1}{2}M{v}^2=\frac{{\left(P_{\mathrm{Nu}}\right)}^2}{2M}=\frac{1}{2M}{\left[\frac{hv}{c}\right]}^2$
Loss in internal energy $=E_{\mathrm{\gamma }}+{\mathrm{KE}}_{\mathrm{Nu}}$
$=hv+\frac{1}{2M}{\left[\frac{hv}{c}\right]}^2$
$=hv\left[1+\frac{hv}{2M{c}^2}\right]$