All I want is a hint, since this is a textbook exercise and I should do it myself.
The problem statement goes something like this:
Let $C_\ell$ be the cycle of length $\ell+1$ (to be excruciatingly specific, this is the $\{R\}$-model of an areflexive, symmetric binary relation $R$ on a set $\{a_0,\dots, a_\ell\}$ of $\ell+1$ elements, $\ell \ge 2$, satisfying $$ a_0Ra_1R\cdots a_\ell R 0 $$ and such that $a_i Ra_j$ for no other $i, j$). For an integer $k$ satisfying $2^k \le \ell$, if $\phi$ is an $R$-sentence with quantifier depth at most $k$, show that $$ C_\ell \models \phi \iff C_\ell \sqcup C_\ell \models \phi. $$
I have tried an induction on $k$, but I haven't been able to get that to work: Peeling off quantifiers will usually turn a sentence into a non-sentence. My primary difficulty is understanding where the "$2^k$" comes from. It is true that any induced subgraph of $C_\ell \sqcup C_\ell$ on at most $\ell/2$ vertices is isomorphic to an induced subgraph of $C_\ell$ (just space out the connected components by dummy vertices - you won't run out!). So why isn't this true with $2k \le \ell$ instead of $2^k \le \ell$?
The best hint I have received was "Ehrenfeucht-Fraisse games", and actually a full solution can be found on pages 84-87 of Moschovakis's informal notes here: https://www.math.ucla.edu/~ynm/lectures/lnl.pdf.