Strong and supercompact cardinals are $\varSigma_2$-reflecting; extendible cardinals are $\varSigma_3$-reflecting. It is of course possible to build larger degrees of correctness into the definitions of large cardinals; if I understand correctly, this is essentially what Joan Bagaria does in his paper "$C^{(n)}$-Cardinals" (Arch. Math. Logic 51 (2012): 213–40; doi: 10.1007/s00153-011-0261-8).

But is there any 'standard' large cardinal notion—basically, one not specifically formulated using the notion of $\varSigma_n$-correctness or something equivalent—such that cardinals of that type are $\varSigma_n$-reflecting for $n > 3$?