The pop is likely due to the voltage on some capacitors taking their time to approach their steady-state voltage. While the cap voltage is transitioning, it will be injecting the voltage into the amplifier, which will be heard at the speaker.
The only thing I can see that would be causing this is the capacitors in the 'equaliser'/tone-control section (all other caps operate at close to 0V DC across them). This can be tested by:
1) reducing the volume control to minimum. If the pop is still occurring, then the preamp has no part in the pop.
2) shorting out R21 (which effectively connects the TDA2030A pin 1 to GND). If the pop stops, then it is the equaliser section causing the issue.
If the above points the problem at the tone section, removing the pop can be done by shorting out R21 on power-up and power-down using perhaps a JFET or two.