不肯(bù kěn) - to be unwilling to, to refuse to, will not in Mandarin

不肯 is a refusal of the will, not of ability. It says the person could do the thing and has decided not to, which is why it so often carries a note of stubbornness or exasperation: 他不肯说 means he is holding out on you.

Contrast it with 不能 (cannot, not permitted) and 不会 (does not know how). Swapping them changes the accusation entirely: 他不肯来 is he will not come, 他不能来 is he is unable to come. English "won't" hides that difference; Chinese does not.

肯 barely appears on its own in modern speech, so treat 不肯 as the everyday form and 愿意 (yuànyì) as the positive counterpart. 他肯来 is possible but reads as slightly bookish; 他愿意来 is the natural way to say he is willing.