Robust Standard Errors for Control Function Approach?
123, the CF method is a specific case of when we have generated regressors. Check the appendix to chapter 6, pages 157-160, for the asymptotic covariance matrix of $N^{1/2}(\hat\beta_{CF}-\beta)$. The expression is truly enormous. ;)

Instead I will just cite wooldridge on page 160, 2nd paragraph:

«If $\mathbf{G}\neq \mathbf{0}$, then (...). Neither the usual 2SLS variance matrix estimator nor the heteroskedasticity-robust form is valid in this case.»The $\mathbf{G}\neq \mathbf{0}$ is the case when we have generated regressors.

Good luck ;)