Closure of regular languages are closed under certain cut operations
Take a DFA $\langle Q,q_0,F,\delta \rangle$ for $L$. For every state $q \in Q$, it is known that
$$
N_q = \{ n : \delta(q,x) \in F \text{ for some $|x|=n$} \}
$$
is an eventually periodic set. In these terms, we have
$$
f(L) = \bigcup_{q \in Q} \{ w : \delta(q_0,w) = q \text{ and } |w|+f(|w|) \in N_q\}.
$$
A few simple arguments show that $f$ is admissible (the set of regular languages is closed under $f$) if for every modulus $m$, there is a DFA $\langle Q',q'_0,\delta' \rangle$ over the alphabet $\{0\}$ which "computes" $|w|+f(|w|) \bmod m$, in the sense that on input $0^n$, you can recover $n+f(n) \bmod{m}$ from $\delta'(q'_0,0^n)$. This, in turn, happens if for every modulus $m$, the function mapping $n$ to $n+f(n) \bmod{m}$ is eventually periodic. Since $n \bmod{m}$ itself is periodic, we conclude that:


  A function $f$ is admissible iff for all $m$, the function $\psi_{f,m}(n) = f(n) \bmod{m}$ is eventually periodic.


The Chinese Remainder Theorem shows that it suffices to consider values of $m$ which are prime powers.

All polynomials are clearly admissible: if $f$ is a polynomial, then $\psi_{f,m}$ has period $m$. Exponential functions $a^n$ are also admissible: if $m=p^k$, then either $p \mid a$, in which case $\psi_{f,m}$ is eventually zero, or $(p,a)=1$, in which case Euler's formula shows that $\psi_{f,m}$ has period $\varphi(m)$.

I'm not sure if the set of admissible functions can be characterized any further – this is an interesting research direction.