40Hz simply is no useful audio frequency for this low power amplifer.
Most probably you connected the 10uF at the GAIN pins.
Then you may expect a cutoff frequency of about 106Hz (here the 150 Ohms dominate)
fc = 1 / ( 2 x Pi x 150 Ohms x 10uF)
(Because of circuit this is no exact calculation)
Also, read the datasheet, specifically "output voltage vs supply voltage". The maximum peak to peak voltage is (roughly) 1V - 2V less than the power supply rail, due to diode and transistor junctions at the output. It is not advertized as a rail-to-rail type of amplifier. 6.2V p-p output with a supply voltage of 8V is exactly in line with the product specification.
There are a lot of audio amplifiers where you are free to set (about) any gain.
The LM86 is somehow special with this.
You say similar.
You need to specify this. Similar in kind of: price, size, power, voltage, current, part count...
I recommend to go to a audio IC manufacturer´s web site and use their selection tool.
3V/10mV= a gain of 300 times. The LM386 is cheap and is not really hifi since with a gain of 200 times its hiss can be heard and will be worse with a gain of 300 times.
Here is a 22 ohm resistor in series with a capacitor to ground to make a gain of 1000 times. Increase the value of the resistor for less gain:
I tried this circuit for 1000 gain in Proteus simulation. For the 10mV(pk-pk) input, i am only getting 100mV(pk-pk) only.
Supply 8V. Gain 1000. Frequency 200 Hz.