Green’s Dictionary of Slang

seek n.

SE in slang uses

In phrases

on the seek

(UK Und.) of a prostitute, walking the streets in search of clients.

[UK]Leaves from Diary of Celebrated Burglar 42/2: She kept a second rate ‘dossing crib,’ and one of the ‘moths’ who lived at her house, ‘picked up’ Charley one night while out on the ‘seek’.