As far as I understand, what you call $f_P$ is usually called monomial symmetric function as is denoted $m_P$. So I interpret your question as asking for an algorithm converting monomial symmetric funtion to power sum symmetric function. Such an algorithm seems to be described here: 

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4656270/

Also there is one which was implemented in [Symmetrica][1] (which is the one used in Sagemath) more than twenty years ago. I don't think the algorithm was documented anywhere but as a comment in the code `tmp.c` . The comment seems to indicate that they use a recursive divide and conquer method using the following recursion step:

 $$
  m_{a_1,a_2,...,a_n,a_{n+1},...a_{2n}} =
   m_{a_1,...,a_n} * m_{a_{n+1},..,a_{2n}} - \text{terms of length} <2n
 $$

You can compute the rest of the term using [this answer][2].

Anyway, unless you really need to say something about the algorithm, I would recommend using a computed algebra system which already has this algorithm implemented. 


  [1]: http://www.algorithm.uni-bayreuth.de/en/research/SYMMETRICA/ "Symmetrica"
  [2]: https://math.stackexchange.com/questions/395842/decomposition-of-products-of-monomial-symmetric-polynomials-into-sums-of-them