Assuming you have $n$ variables then for $k\geq n$, Robin Chapman's identity above can be written as $$(p_n,p_{n-1},\dots, p_1)\begin{pmatrix} e_1 & 1 & \cdots & 0 \\\ -e_2 & 0 & \ddots & \vdots \\\ \vdots & \vdots & \ddots & 1 \\\ (-1)^{n-1}e_n & 0 & \cdots & 0 \end{pmatrix}^{k-n}=(p_k,p_{k-1},\dots, p_{k-n+1})$$
Now to finish the job you need to express the $e_i$'s in terms of the power sum symmetric functions too. This is given by $$e_n=\sum_{|\lambda|=n}(-1)^{|\lambda|-l(\lambda)} z_{\lambda}^{-1}p_{\lambda}$$ where $|\lambda|$ is the size of the partition $\lambda$ and $l(\lambda)$ is its length, $p_{\lambda}=p_{\lambda_1}p_{\lambda_2}\cdots$ and $$z_{\lambda}=\prod_{i\geq 1}\left(i^{m_i}\cdot m_i!\right)$$ where $m_i$ is the number of parts of $\lambda$ equal to $i$.
I thought I'd remark that the formulas you are quoting are all valid in $\Lambda_{\mathbb{Q}}$, the ring of symmetric functions in infinitely many variables while the one you are searching for is not, because the $p_\lambda$'s are an orthogonal basis in this ring with $\langle p_{\lambda},p_{\mu}\rangle =\delta_{\lambda \mu}z_{\lambda}$. This is also the same reason why the formula for Schur polynomials may contain arbitrary $p_{\lambda}$'s in it. In fact the reason why that formula is important is because it gives you the transition from the basis of Schur polynomials to that of power sum polynomials in $\Lambda_{\mathbb{Q}}$.