I'm guessing that the headphone jack does not have a mute circuit when turning the amp on/off.
If so than what you are hearing are the noises that amp's typically make during turn on/off but which we are spared from hearing (through the speakers) as most amps mute the speaker outs. The clicking sound that most amps have after a few seconds of being on {or immediately upon turning the amp off) is the sound of the relay unmuting or muting the speaker outputs.
If so than what you are hearing are the noises that amp's typically make during turn on/off but which we are spared from hearing (through the speakers) as most amps mute the speaker outs. The clicking sound that most amps have after a few seconds of being on {or immediately upon turning the amp off) is the sound of the relay unmuting or muting the speaker outputs.