As an additional mass $m$ is added, the spring is further stretched by a distance $X$ , therefore

$kX=mg$

$\Rightarrow k=\frac{mg}{X}$

The time period of the spring-mass system when both the masses $\left(m+M\right)$ are suspended from the spring,

$T=2\pi \sqrt{\frac{(M+m)}{k}}$

$\Rightarrow T=2\pi \sqrt{\frac{\left(m+M\right)X}{mg}}$