How can I modify the charging circuit of the solar powered system using super-capacitor with or without battery (diagram is attached)?
Your main problem is that the panel is a poor match for the rest of the system. It's killing the battery by overcharging it.

The lead-acid battery functions as a voltage regulator; you can't force the voltage drop across it to more than about 2.4 V per cell. This means that your solar panel will be forcing about 30 mA into the battery, even when it is already fully charged. You really need to add a circuit that limits the float-charge voltage going to the battery.

This is just as important if you add or switch to supercapacitors as your energy storage device. Supercapacitors have very strict voltage limits, and if you exceed them, you will severely shorten their lifetime. Also, they have less ability to equalize the voltage among the individual series-connected devices, so you'd need to add circuitry to handle that, too.