fermions - Can we treat $psi^{c}$ as a field independent from $psi$?

When we derive the Dirac equation from the Lagrangian, $$ \mathcal{L}=\overline{\psi}i\gamma^{\mu}\partial_{\mu}\psi-m\overline{\psi}\psi, $$ we assume $\psi$ and $\overline{\psi}=\psi^{*^{T}}\gamma^{0}$ are independent. So when we take the derivative of the Lagrangian with respect to $\overline{\psi}$, we get the Dirac equation $$ 0=\partial_{\mu}\frac{\partial\mathcal{L}}{\partial\left(\partial_{\mu}\overline{\psi}\right)}=\frac{\partial\mathcal{L}}{\partial\overline{\psi}}=\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi. $$

Now if we include a term with charge conjugation, $\psi^{c}=-i\gamma^{2}\psi^{*}$, into the Lagrangian (like $\Delta\mathcal{L}=\overline{\psi}\psi^{c}$), does this $\psi^c$ depend on $\overline{\psi}$ or $\psi$? Why or why not?

If $\psi^{c}$ depends on $\psi$, why wouldn't the reason that $\overline{\psi}$ and $\psi$ are independent apply for $\psi^{c}$ and $\psi$?

If $\psi^{c}$ depends on $\overline{\psi}$, how should we take derivative of $\Delta\mathcal{L}$ with respect to $\overline{\psi}$?