  In this paper we introduce and analyze an iteratively re-weighted algorithm,
that allows to approximate the weak solution of the $p$-Poisson problem for $1
< p \leq 2$ by iteratively solving a sequence of linear elliptic problems. The
algorithm can be interpreted as a relaxed Ka{\v c}anov iteration, as so-called
in the specific literature of the numerical solution of quasi-linear equations.
The main contribution of the paper is proving that the algorithm converges at
least with an algebraic rate.
