(You are considering the uniform distribution on a discrete simplex. I'm not aware of specific results in the literature, and a brief internet search didn't reveal anything.)

A simple way is to use the joint (probability) generating function of $X_1,\ldots,X_n$.

It is easy to see that it can be writen as $$\mathbb{E}t_1^{X_1}\ldots t_n^{X_n}=\frac{1}{{n+k-1 \choose k}}[x^k] \prod_{i=1}^n \frac{1}{1-t_ix}$$ Using that one finds e.g. that for $j=(j_1,\ldots,j_n)$ and for the polynomials $p_j(X_1,\ldots,X_n)={X_1 \choose j_1}{X_2 \choose j_2}\cdots {X_n \choose j_n}$ (product of binomial coefficients) and $s=j_1+\ldots+j_n$ $$\mathbb{E} p_j(X_1,\ldots,X_n)=\frac{{k \choose s }}{{n+s-1 \choose s}}$$

(You are considering the uniform distribution on a discrete simplex. I'm not aware of specific results in the literature, and a brief internet search didn't reveal anything.)

A simple way is to use the joint (probability) generating function of $X_1,\ldots,X_n$.

It is easy to see that it can be writen as $$\mathbb{E}t_1^{X_1}\ldots t_n^{X_n}=\frac{1}{{n+k-1 \choose k}}[x^k] \prod_{i=1}^n \frac{1}{1-t_ix}$$ Using that one finds e.g. that for $j=(j_1,\ldots,j_n)$ and for the polynomials $p_j(X_1,\ldots,X_n)={X_1 \choose j_1}{X_2 \choose j_2}\cdots {X_n \choose j_n}$ (product of binomial coefficients) and $s=j_1+\ldots+j_n$ $$\mathbb{E} p_j(X_1,\ldots,X_n)=\frac{{k \choose s }}{{n+s-1 \choose s}}$$

ADDED: the first appearance of moments seems to be in Whitworth, Choice and Chance. Withe 1000 exercises, 5th edition (1901), exercise 945 (bottom of page 323). See https://archive.org/details/choicechancewith00whituoft/page/n4

(You are considering the uniform distribution on a discrete simplex. I'm not aware of specific results in the literature, and a brief internet search didn't reveal anything.)

A simple way is to use the joint (probability) generating function of $X_1,\ldots,X_n$.

It is easy to see that it can be writen as $$\mathbb{E}t_1^{X_1}\ldots t_n^{X_n}=\frac{1}{{n+k-1 \choose n}}[x^n] \prod_{i=1}^n \frac{1}{1-t_ix}$$ Using that one finds e.g. that for $j=(j_1,\ldots,j_n)$ and for the polynomials $p_j(X_1,\ldots,X_n)={X_1 \choose j_1}{X_2 \choose j_2}\cdots {X_n \choose j_n}$ (product of binomial coefficients) and $s=j_1+\ldots+j_n$ $$\mathbb{E} p_j(X_1,\ldots,X_n)=\frac{{k \choose s }}{{n+s-1 \choose s}}$$

(You are considering the uniform distribution on a discrete simplex. I'm not aware of specific results in the literature, and a brief internet search didn't reveal anything.)

A simple way is to use the joint (probability) generating function of $X_1,\ldots,X_n$.

It is easy to see that it can be writen as $$\mathbb{E}t_1^{X_1}\ldots t_n^{X_n}=\frac{1}{{n+k-1 \choose k}}[x^k] \prod_{i=1}^n \frac{1}{1-t_ix}$$ Using that one finds e.g. that for $j=(j_1,\ldots,j_n)$ and for the polynomials $p_j(X_1,\ldots,X_n)={X_1 \choose j_1}{X_2 \choose j_2}\cdots {X_n \choose j_n}$ (product of binomial coefficients) and $s=j_1+\ldots+j_n$ $$\mathbb{E} p_j(X_1,\ldots,X_n)=\frac{{k \choose s }}{{n+s-1 \choose s}}$$