LM386 audio amplifier with push-pull output
The circuit should be ok if you fix the polarity on Q3.  Electrolytic caps conduct if they are back biased, so running C2 and C3 back to back is ok to drive Q1 base, and C4 is biased properly to drive the speaker.  1 W is overkill for the resistors, as is 4 W on the pot.  You can use regular 1/4 W or 1/8 W resistors everywhere.  Add some bypass capacitance on the LM386.  You might want to try it without the LM386 - you have a lot of gain in both stages.  The voltage swing required on your class AB input is only millivolts.