Peptide nucleic acid (PNA)-neocuproine conjugates have been shown to efficiently catalyse the cleavage of RNA target sequences in the presence of Cu2+ ions in a site-specific manner. These artificial enzymes are designed to force the formation of a bulge in the RNA target, the sequence of which has been shown to be key to the catalytic activity. Here, we present a further investigation into the action of Cu2+-dependent PNAzymes with respect to the dependence on bulge composition in 3- and 4-nucleotide bulge systems. Cu2+-dependent PNAzymes were shown to have a clear preference for 4-nucleotide bulges, as the cleavage of 3-nucleotide bulge-forming RNA sequences was significantly slower, which is illustrated by a shift in the half-lives from approximately 30 min to 24 h. Nonetheless, the nucleotide preferences at different positions in the bulge displayed similar trends in both systems. Moreover, the cleavage site was probed by introducing critical chemical modifications to one of the cleavage site nucleotides of the fastest cleaved 4-nucleotide RNA bulge. Namely, the exclusion of the exocyclic amine of the central adenine and the replacement of the 2′-hydroxyl nucleophile with 2′-H or 2′-OMe substituents in the RNA severely diminished the rate of RNA cleavage by the Cu2+-dependent PNAzyme, giving insight into the mechanism of cleavage. Moreover, the shorter recognition arm of the RNA/PNAzyme complex was modified by extending the PNAzyme by two additional nucleobases. The new PNAzyme was able to efficiently promote the cleavage of RNA when fully hybridised to a longer RNA target and even outperform the previous fastest PNAzyme. The improvement was demonstrated in cleavage studies with stoichiometric amounts of either PNAzyme present, and the extended PNAzyme was also shown to give turnover with a 10-fold excess of the RNA target.