How to power a widget, while the li-ion battery is charging?
I've thought about this requirement recently.

As long as you keep the current from Vin to load partitioned away from the battery current then you should have minimal problems. ie if current went from Vin to battery and then Battery to load, the system needs to do extra work to be able to ignore Iin.  

ie battery charge proper needs to be able to deal with net battery current during charging. As long as you make it possible  for it to manage battery current during charging all will be well.  This way (digram below) the battery-controller does not "see: what the external source is doing. 


Iin -> Regulator -> Battery -> Battery Controller -> Load      x


BUT if in goes to load directly and battery connects to load independently, then all is kept partitioned.


Iin -> Regulator -> Load
Battery - battery controller - Load


In practice this may be as simple as putting the load on the Vcharge side of the battery current sense. Current sense then sees true battery current (in or out) and is "unaware" of actual load current. 

SO

If the external supply can operate the widget directly it will run without battery involvement. If not (say 100 mA charge, 300 mA load) then it will draw battery current to make up the difference. 

End of constant current phase is tripped when the battery voltage reaches Vmaxchg *=(usually 4.2V/cell). This voltage may be slightly affected by available charge current but not much. [[Note: Charge % during constant  current phase is usually rather less than 90%]]. So end of CC phase will not be much affected by widget running.

End of CV (constant voltage) phases occurs when Ichg drps to some % of Imax.
 If external supply cannot support widget plus Ichg_current directly then Ichg wll decrease and  charging may be terminated. You could either disable charge termination during widget running time or add a fake current value into the termination control circuitry to make it look like Ichg is steady when the widget runs. Easier done than thought of :-). 

The subterfuge of adjusting apparent charge rate does no harm as battery will not be charging during these periods (and normal operation assumes when the widegt stops operating.