LM386 amp circuit lag
It is probably charge time of C5.  C5 must reach full charge before the DC level of the amplifier is settled down.  But C5 is charging through some large resistances, so it takes some time.

C5 is charging through about 70k of resistance.  Resistors add in series, so R2 is in series with the input resistance of the LM386 (about 50k) and R3 is 10K series resistance on the other side.

The charge time is proportional to the RC product.  Input RC is (70k * 10μ) or 0.7 seconds.

If C5 is too low a value, the amplifier sound will be tinny.  If it is too large, you will get the long turn-on transient that you are seeing.

All to say, try reducing C5 to 0.1μF or 0.047μF.