According to the image, I am getting good amplified sound form my circuit. but when I am trying to give the a single supply source, the circuit is not working. The speaker goes to absolute silence. Someone please figure this out.

 

Just a guess but your drawing implies 2 different power supplies. Have you connected the grounds from the two supplies?

 

You have shorted one of the ISD1820 speaker outputs to ground. Below is the diagram from the datasheet of how to connect it to an LM386.