How to invert an arithmetic function where Möbius inversion may not apply?

Question

I'm playing around with problems in order to gain a very basic insight into number theory, and I am looking at the process of inversion. Take a convergent arithmetic function of the general form

$$f(x)=\lim \limits_{k \to \infty}\biggl(\sum_{n=1}^kg\bigl(h_n(x)\bigr)\biggr)^k$$

where $h_n$ is some arbitrary $n$-dependent arithmetic function on $x$.

I want to find $g$ in terms of $f$.

My first thought was Möbius inversion (mainly because that's what I happen to be exploring at the moment). But this raises three potential issues:

1. $\sum_{n=1}^k$ ranges across all natural numbers $n\le k$, not just divisors of $k$. Is there a way round this? (Maybe something to do with prime powers and Dirichlet convolution...?)
2. How to deal with $\lim \limits_{k \to \infty}$? It makes no sense to substitute $k=\infty$ because of the $k$th power term.
3. Is it possible to apply Möbius inversion to a function raised to the $k$th power?

I'm learning as I go here, so please accept that I may need a little more help than other people.