To find $E\left[X^{2}\right]$, we calculate:
$$
E\left[X^{2}\right]=\sum_{x=1}^{4} x^{2} \cdot f(x)
$$
Substituting the values:
$$
\begin{aligned}
E\left[X^{2}\right] & =1^{2} \cdot \frac{9}{24}+2^{2} \cdot \frac{7}{24}+3^{2} \cdot \frac{5}{24}+4^{2} \cdot \frac{3}{24} \\
& =1 \cdot \frac{9}{24}+4 \cdot \frac{7}{24}+9 \cdot \frac{5}{24}+16 \cdot \frac{3}{24} \\
& =\frac{9}{24}+\frac{28}{24}+\frac{45}{24}+\frac{48}{24} \\
& =\frac{9+28+45+48}{24} \\
& =\frac{130}{24}
\end{aligned}
$$
So, the correct answer is:
C. $\frac{130}{24}$
