Specifically catalyzes the deamidation of the C-terminal glutamine of the prokaryotic ubiquitin-like protein Pup to glutamate, thereby rendering Pup competent for conjugation. Probably also displays depupylase (DPUP) activity, removing conjugated Pup from target proteins; thus may be involved in the recycling of Pup and may function similarly to deubiquitinases (DUBs) in eukaryotes to prevent or promote proteasomal degradation of certain proteins. [prokaryotic ubiquitin-like protein]-C-terminal-L-glutamine + H2O = [prokaryotic ubiquitin-like protein]-C-terminal-L-glutamate + NH4(+) ATP is required for the deamidation reaction but is not hydrolyzed during this reaction. Protein degradation; proteasomal Pup-dependent pathway. Pupylation is severely impaired and the steady-state levels of the two known proteasomal substrates FabD and PanB are drastically increased. Is the only enzyme acting as a deamidase of Pup during pupylation. Belongs to the Pup ligase/Pup deamidase family. Pup deamidase subfamily.