Green’s Dictionary of Slang

1. deceit; an illusion, a disguise.

1819		Observer (London) 21 Mar. 4/1: He says he’s out of condition. This may be a ‘throw-off’.
1903		A.H. Lewis Boss 173: Did you ever see such a throw-off?
1933		C.W. Willemse A Cop Remembers 20: I am satisfied that job was the work of people who knew a great deal about the movements of the Lindbergh family. Was it done for revenge, perhaps, with a throw off on the ransom note?
1938		D. Runyon ‘The Brakeman’s Daughter’ in Runyon on Broadway (1954) 502: This smile is strictly a throw-off, and Big False Face is often smiling when he is by no means amused.

2. a hostile or critical remark or allusion.

1940

F. Hunt Long Trail from Texas 39: You dad-bern [sic], Arkansas thowoffs! I’ll larn ya yer manners afore ya ever wet yer feet ag’in.

3. (US Und.) a supposedly legitimate business which in fact masks a criminal one.

1908

‘Old Sleuth’ Dock Rats of N.Y. (2006) 50: The men ostensibly were fishermen, and their boat was stated to be a fishing-boat; and to lend color to the claim, the men did go off between times on fishing expeditions, and the latter little trick had been their best ‘blind’ and ‘throw off.’.