For small $n$ you could just evaluate this by explicit calculation. For large $n$ there is a useful asymptotics:

Wigner [1] showed that, under quite general conditions on the distribution of the independent matrix elements, and setting their variance equal to unity, the large-$n$ asymptotics of the expectation value of $\text{Tr}\,(X^k)$ is given by a Catalan number, $$\frac{1}{n}E\,[\text{Tr}\,(X^k)]=\frac{1}{s+1}{{2s}\choose{s}}+O(n^{-2}),\;\;k=2s\;\;\text{even}.$$ The distribution of this trace satisfies a central-limit-theorem, converging for large $n$ to a Gaussian distribution with variance $1/\pi$. See for example Central limit theorem for traces of large random symmetric matrices with independent matrix elements.

[1] E.P. Wigner, On the Distribution of the Roots of Certain Symmetric Matrices (1958).

For small $n$ you could just evaluate this by explicit calculation. For large $n$ there is a useful asymptotics:

Wigner [1] showed that, under quite general conditions on the distribution of the independent matrix elements, and setting their variance equal to unity, the large-$n$ asymptotics of the expectation value of $\text{Tr}\,(X^k)$ is given by a Catalan number, $$E\,[\text{Tr}\,(X^k)]=\frac{n}{s+1}{{2s}\choose{s}}+O(n^{-1}),\;\;k=2s\;\;\text{even}.$$ The distribution of this trace satisfies a central-limit-theorem, converging for large $n$ to a Gaussian distribution with variance $1/\pi$. See for example Central limit theorem for traces of large random symmetric matrices with independent matrix elements.

[1] E.P. Wigner, On the Distribution of the Roots of Certain Symmetric Matrices (1958).

For small $n$ you could just evaluate this by explicit calculation. For large $n$ there is a useful asymptotics:

Wigner showed that, under quite general conditions on the distribution of the independent matrix elements, and setting their variance equal to unity, the large-$n$ asymptotics of the expectation value of $\text{Tr}\,(X^k)$ is given by a Catalan number, $$\frac{1}{n}E\,[\text{Tr}\,(X^k)]=\frac{1}{s+1}{{2s}\choose{s}}+O(n^{-2}),\;\;k=2s\;\;\text{even}.$$ The distribution of this trace satisfies a central-limit-theorem, converging for large $n$ to a Gaussian distribution with variance $1/\pi$. See for example Central limit theorem for traces of large random symmetric matrices with independent matrix elements.

For small $n$ you could just evaluate this by explicit calculation. For large $n$ there is a useful asymptotics:

Wigner [1] showed that, under quite general conditions on the distribution of the independent matrix elements, and setting their variance equal to unity, the large-$n$ asymptotics of the expectation value of $\text{Tr}\,(X^k)$ is given by a Catalan number, $$\frac{1}{n}E\,[\text{Tr}\,(X^k)]=\frac{1}{s+1}{{2s}\choose{s}}+O(n^{-2}),\;\;k=2s\;\;\text{even}.$$ The distribution of this trace satisfies a central-limit-theorem, converging for large $n$ to a Gaussian distribution with variance $1/\pi$. See for example Central limit theorem for traces of large random symmetric matrices with independent matrix elements.

[1] E.P. Wigner, On the Distribution of the Roots of Certain Symmetric Matrices (1958).