As drawn, there is no limit on the output voltage, without a battery. C2 will explode.
I question the placement of the LED.
I added a Zener diode to limit the voltage. Mayve 6.3V or even 12V, something!
Did you design this circuit?
Do you want the LED to come on when the battery is charged, or do you want to see the charging current in the LED?
It could be that the LED+resistor works like the Zener to limit the voltage, but the LED is in the wrong place.
View attachment 145591
The charger will work just fine without C2. (assuming there is something to limit the voltage) Some battery chargers do 120 (or 100) pulses/second charging. Pulse charging is just fine.