Hi Forum Members,
Many thanks to all of you for giving me the suggestions.
The Following Tests were done:

Test1:
Observed for change in frequency on CLKOUTPIN (CLKOUT is giving 32kHz) using a CRO, when the Mains Power is switched OFF and the PCF8563 gets supply from Battery.
Observed1: There is no change in the frequency when the Mains Power is switched OFF.
Result1:
The (in) accuracy still varies when the Power is swicthed OFF and Switched ON.


Test2:
As Micheal suggested, we changed the RTC Reading Routine to read the RTC in IIC(I2C) continuous mode.
Observed2:
There was no change in the accuracy of the clock, whether the mains power is ON or OFF.
Result2:
This method worked perfectly OK and now the RTC is very accuracte right to the second even after 12 days+nights.

A Big Thank You to Micheal Karas, and this forum for their suggestions.

I request the Forum moderator to close the thread as the problem of RTC-inaccuracy is now solved.

Bye,
Mr. Kiran V. Sutar.