I assume your trade $V(S,K,t,T)$ is European. Its payoff is:
$$\begin{align}
V(S,K,T,T)&=C^2(S,K,T,T)
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&=\max(S_T-K,0)^2
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&=\boldsymbol{1}_{\{S_T\geq K\}}(S_T-K)^2
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&=\boldsymbol{1}_{\{S_T\geq K\}}f(S_T)
\end{align}$$
where $f(x)=(x-K)^2$. By Carr-Madan's static replication formula (see this question or this paper), we have(1):
$$\begin{align}
f(S_T)&=f(K)+f'(K)(S_T-K)+\int_0^{K}f''(k)(k-S_T)^+\text{d}k+\int_{K}^{\infty}f''(k)(S_T-k)^+\text{d}k
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&=2\int_0^{K}(k-S_T)^+\text{d}k+2\int_{K}^{\infty}(S_T-k)^+\text{d}k
\end{align}$$
where $(x)^+=\max(x,0)$. Multiplying by $\boldsymbol{1}_{\{S_T\geq K\}}$:
$$\boldsymbol{1}_{\{S_T\geq K\}}f(S_T)=2\int_{K}^{\infty}(S_T-k)^+\text{d}k$$
Multiplying by the discount factor $D(t,T)$ and taking the conditional expectation under the risk-neutral measure $Q$, we get the following theoretical replicating strategy:
$$V(S,K,t,T)=2\int_{K}^{\infty}C(S,k,t,T)\text{d}k$$
Given in practice there is no availability of a continuum of call options, the following approximation is made:
$$V(S,K,t,T)\approx2\sum_{i=0}^nC(S,k_i,t,T)\delta_i$$
where $\{k_i:i=0,\dots,n\}$ are the quoted strikes with $k_0=K$ and $\delta_i=k_{i+1}-k_i$.
(1) We have chosen as threshold value the strike $K$.