to answer why add capacitor:
I have buck-boost switching regulator that regulates voltage to 3.3V, I also have some loads coming on/off, beeper, LEDs, causing voltage drop on the battery so in my tests placing 220uF cap reduced voltage drop, it does not affect it much when batteries are fresh but somewhere along the way bulk capacitor helped, so I decided to stick to it.
to answer about mixing batteries:
first time when it happened, device was working steady discharging predictable for few days, then suddenly stopped and that's when I measured voltage and all 3 out of 4 batts were 1ish volt and one was waaay below that.
second time pretty much same story, device prematurely quit working and same measurement, again, one battery out of 4 quit
I will get different brand and will see how it perform, if you saying that inrush current won't be too large to damage batteries and and it is possible that one battery is faulty to do trick like that I suspect and hope that was the case.
I want to add bulk 220uF tantalum polymer (for smaller size) capacitor across batteries terminals
Why? What possible reason would you need a capacitor in parallel with the batteries?
my question is: when I power device on (I apologize for lack of proper terminology) I think capacitor will be sucking alot of current form batteries for some time, almost shorting batteries for little while, which per my understanding is not very healthy for battery, right?
The cap will indeed suck as much charge as it needs when first connected to the battries. If the batteries are in good shape, they should easily be able to supply the current without any dip in voltage.
No, you do not need a current limiting resistor.
The real question as I have already asked is why you want this capacitor?
apparently this 300mV battery is pooping everything else, what may have caused this battery operate normal and then suddenly drop so low
Did you mix old and new batteries?
There is no reason I can think of that 1 battery will suddenly drop in volts unless one battery is already partly drained or came faulty in the pack.