Green’s Dictionary of Slang

1. a magistrate [= beak n.¹ (1)].

1892

‘Doss Chiderdoss’ ‘The Rhyme of the Rusher’ in Sporting Times 29 Oct. n.p.: The once a week made me go ta-ta / For a month on the can’t keep still.

2. cheek.

1925		(con. WWI) Fraser & Gibbons Soldier and Sailor Words 214: Once A Week: Cheek.
1938		F.D. Sharpe Sharpe of the Flying Squad 332: ‘He’s got some once a week.’ – ‘He is cheeky.’.
a.1945		L. Payne private coll. n.p.: Cheek Once a Week.
1960		J. Franklyn Dict. of Rhy. Sl.
1977		D. Powis Signs of Crime 195: Once a week Cheeky in the sense of impertinence: ‘He’s got plenty of once a week.’.