Green’s Dictionary of Slang

clarinet-player n.

[pun on SE, the key image is that of blowing]

(Aus.) a fellator or fellatrix.

H.N. Carey Sl. Venery I 45: clarinet player—A cock sucker.
[Aus]‘No 35’ Argot in Simes DAUS 46/1: clarinet player One who commits fellatio.
[UK]P. Theroux Picture Palace 314: Doris with a grip on Kenny’s joystick; Kenny nibbling; Doris tooting his clarinet.