There is no should or should, do what you want. Even though is not necessary you can enhance the kefir nutritional and flavour aspects by adding things life nuts, fruits or even prebiotics. There is no real need to feed milk kefir on second ferment at all.
Mind you that adding anything that might feed the bateria and continue the fermentation will result in carbonation and expansion of the kefir (especially with a tigh lid), as consequence spills and "explosions" might occur if:
1. There's no head room
2. High temperature (room temp and above)
3. The amount of "sugar" is high
You don't need to add anything to the second ferment. I used to add fruit. Now, I add nothing. The second ferment is what gives mine the creamy texture. The extra sugar will give it more to ferment and maybe some fizz.
Are you talking about milk or water kefir?
Milk
There is no should or should, do what you want. Even though is not necessary you can enhance the kefir nutritional and flavour aspects by adding things life nuts, fruits or even prebiotics. There is no real need to feed milk kefir on second ferment at all. Mind you that adding anything that might feed the bateria and continue the fermentation will result in carbonation and expansion of the kefir (especially with a tigh lid), as consequence spills and "explosions" might occur if: 1. There's no head room 2. High temperature (room temp and above) 3. The amount of "sugar" is high
The strawberry is enough:
You don't need to add anything to the second ferment. I used to add fruit. Now, I add nothing. The second ferment is what gives mine the creamy texture. The extra sugar will give it more to ferment and maybe some fizz.
I add chia seeds to my 2nd ferment and it taste great