I have a quick clarification question: When you say "Suppose $Q(X_1,\ldots,X_n)$ is in $I^t$", you mean suppose there is are $\beta_1,\ldots,\beta_n\in R$ such that $Q(\beta_1,\ldots,\beta_n)\in I^t$? And then every time you write $\frac{dQ}{dX_1}$, you mean it is evaluated at the $\beta_1,\ldots,\beta_n$? If that's the case, then I interpret the mod $I^{t+1}$ solution as $\beta_1 + a_1,\ldots,\beta_n+a_n$. Thanks!

Thanks David! This will take me a little while to parse, so i might add follow up questions in the comments. But for now I accept this!

Thanks for pointing this out Ryan. In this case, I have a specific $d$ in mind: if $g(c)$ has degree greater than $kd$, then I do not wish to consider such a $c$. The bound I'm seeking on $|C|$ may depend on $d$. I will edit the problem statement to make this clearer.