Are White's Robust standard errors robust to clustered errors? I want to ask about OLS White's 1980 "robust" standard errors. 
The key assumption, is that regression errors $u_i$ have distinct variances $σ_i^2$. Then the variance matrix is: $$\Sigma = \operatorname{diag}(\sigma_1^2, \ldots, \sigma_n^2)$$ with its White's estimator: $$\hat\sigma_i^2 = \hat u_i^2$$ This is the HCE (heteroscedasticity-consistent estimator). 
 Is the White's robust variance of $\hat{\beta}$ estimated by OLS assuming  independence? For example, assume that there is some $i,j$ such that $$Cov(u_i,u_j)\neq0$$ Because $i,j$ are part of a cluster. In that case $\Sigma$ is not diagonal but has the same diagonal as before. Will the White estimator converge to the diagonal of the true $\Sigma$?
No. White's robust errors are only robust to errors that vary linearly based on X(t).  If you have a type of serial correlation or clustering you will want to use Hansen method to adjust standard errors.