1) to ask for in addition
2) to approach one with supplications
3) to ask alms
Part of Speech: verb
From G4314 and G154; to ask repeatedly (importune), that is, solicit: - beg.
These two sites give similar information, with the definition from several dictionaries and statistics on the use of the word.
Perseus Digital Library
University of Chicago's Logion lexicon
Thayer's dictionary plus other information.
From this site's dictionary (in Italian)
Click on the first column to search for that word as a form of the root προσαιτέω; click on the second column to search for that grammatical form of the root προσαιτέω; click on the third column to search for that word and grammatical form; click on the total to search for the root προσαιτέω.