I was wondering if, when running a regression on panel data, clustered standard errors are already correcting for heteroskedasticity. Actually, I have run such a regression and detected heteroskedasticity. Since I used the pooled OLS model I have to cluster the standard errors anyway. Hence, I was hoping that I can address both issues simultaneously. Is that right?
Answering you question: Cluster Robust is also Heteroskedastic Consistent.
I would recommend that you read the A Practitioner's Guide to Cluster-Robust Inference which is a nice piece from Colin Cameron on several aspects of clustered/heteroskedastic robust errors.
Page 20 onward should help you out.