Green’s Dictionary of Slang

pogey n.1

also poggie, pogie, pogy
[? poke n.2 (2), i.e. something in which one is ‘put away’; dial. poghole, a boggy hole or Fr. poche, a pocket]

1. [late 19C–1960s] (US tramp) a workhouse; a relief centre.

2. [1920s–70s] a prison hospital.

3. [1930s–60s] a house of correction, a prison.