Green’s Dictionary of Slang

moan n.

(orig. milit.) a grievance, a complaint.

1911		‘Guns’ & ‘Theeluker’ Middle Watch Musings 12: ‘Guard and Steerage ’ammicks, Sir!’ I wake up with a groan; Why can’t I sleep till 7 a.m.? Once more I had a moan.
1935		A.J. Pollock Und. Speaks n.p.: Moans and Groans, poor luck in gambling or in love; misery.
1956		‘Charles Raven’ Und. Nights 181: To hear the moan they put up, you’d have thought they were being made to sew mailbags at cut price rates.

In phrases

put on the moan (v.)

(US) to complain.

1938

D. Runyon ‘Madame La Gimp’ in Runyon on Broadway (1954) 239: She comes shuffling along putting on the moan about her tough luck.