Express the weighted power sums

Question

Fix $m:=(m_1\ldots,m_k)\in\mathbb{N}^k$ be a sequence of natural numbers and for $i\in\mathbb{N}$ denote by $p_{i,m}:=m_1X_1^i+\dots+ m_k X_k^i$ - a weighted power sum polynomial. It seems that the first $k$ of these are algebraically independent - similarly to the Newton sums. Is there a formula to express the higher weighted power sums in terms of these first $k$?