In this paper, we derive limit laws for the empirical spectral distributions of random band and block matrices with correlated entries. In the first part of the paper, we study band matrices with approximately uncorrelated entries. We strengthen previously obtained results while requiring weaker assumptions, which is made possible by a refined application of the method of moments. In the second part of the paper, we introduce a new two-layered correlation structure we call SSB-HKW correlated, which enables the study of structured random matrices with correlated entries. Our results include semicircle laws in probability and almost surely, but we also obtain other limiting spectral distributions depending on the conditions. Simple necessary and sufficient conditions for the limit law to be the semicircle are provided. Our findings strengthen and extend many results already known.