Yes, there are closed-form expressions. The number $S(n,k)$ you are looking for is the number of weighted integer compositions with weighting function $f(a)=a^p$. Many recursions and other representations for this number exist.
For example, the number $S(n,k)$ from above is precisely the number $d_{S,f}(n,k)$ defined in Eger (2013), Restricted weighted integer compositions and extended binomial coefficients (for $S=\{1,2,3,…,\}$ and $f(a)=a^p$). This paper says that $S(n,k)$ is an extended binomial coefficient, and gives various representations of the extended binomial coefficients. Other relevant literature would be Fahssi (2012), The polynomial triangles revisited, and Shapcott (2013), C-color compositions and palindromes. More relevant literature can be found in the references of these works. Other work of C. Shapcott also addresses part-products of integer compositions, which is related to your case $p=1$.