Polynomial equations for additive functions II

In this sequence of work we investigate polynomial equations of additive functions. We consider the solutions of equation $$\sum _{i=1}^n{f}_i(x^{p_i})g_i(x{)}^{q_i}=0(x\in \mathbb{F}),$$ where $n$ is a positive integer, $\mathbb{F}\subset \mathbb{C}$ is a field, $f_i,g_i\text{\hspace{0.17em}⁣}:\mathbb{F}\to \mathbb{C}$ are additive functions and $p_i,q_i$ are positive integers for all $i=1,\dots ,n$. Using the theory of decomposable functions we describe the solutions as compositions of higher order derivations and field homomorphisms. In many cases we also give a tight upper bound for the order of the involved derivations. Moreover, we present the full description of the solutions in some important special cases, too.