Ok. I avoided this thread because lately, I'm not really "feeling" it. But also because there is no real solution to a POOR or BAD design.
Cheap boomboxes will do that. Especially higher-power cheap ones. Ever turn on a pocket radio -- you know, the ones with 1-1/2" speakers? Does that pop every time you turn it on. But since the volume is so low, it's not really offensive. But with a boombox that outputs wpc instead of milliwatts and 6 or 8" woofers -- that pop is much more noticeable.
Anyhow, high quality boomboxes include timing delay and soft start circuitry to prevent or minimize the pops. If the pop is caused by power transients from the switch action, then installing a small .01uf or so high-voltage capacitor across the switch contacts should help with that. But ideally, the turn on sequence should be source, preamp, then amp and power down sequence should be amp, preamp, source. Of course if you were designing boomboxes on the cheap, you overlook and omit these litle details. It's no coincidence that on quality boomboxes, the innards are jam packed with circuitry whereas on the big black boxes, there is one single small board. All that extra circuitry is not there simply because the manufacturer had money to throw away. Therefore, the pop, is probably normal for those models as they lack the circuitry to supress the pop. On models using IC amps with a mute or remote turn on feature, you can delay the turn on of the amp by cutting a series resistor into that circuit to delay the turn on of the amp. I say live with it. It's the nature of the primitive beast.