I would first check the battery backup circuit. It should not allow the supply voltage to drop below about 2V.
You can also try a larger cap on the supply pins of the PCF8583.
I always use this circuit for power supply of PCF8583 and I never have problem. Diodes should be schotkey, capacitors eg. 100uF (electrolitic) and 100nF (ceramic).
most reading problems solved, but still this problem occurs time to time.
then i changed the part pcf8583, now i'm using ds1307.
with ds1307 everything goes properly. no problem since 10 days.