Can a 3.2V, 400mAh battery charge a smartphone?
Yes, but most smart phones want 5V to charge, so you'll need to boost the voltage.

DC boost convertes are available online at good prices.

do a search on "USB 5V boost" prices start at less than a dollar.

14430 cell pictured does not have match the energy capacity of most cellphones, so you may need several of those cells to do a full charge.
Many powerbanks are based on the larger 18650 cell.