I think that we all struggle to do and say the right things in this situation; but, speak to anyone who has a terminal illness or has just been bereaved, and they say that the worst thing is when people pass by on the other side and avoid contact.
Does this man have other family to support him? Did you hear of his illness through one of them or some other way?
I think that you should visit and just say, in a matter-of-fact way, that you are sorry to hear of his illness and please tell you if there is any sort of help that he needs. Take your cue from him then - if he wants to talk about it he will, and if not he will quickly get on to other subjects. If he is, as you say, a private person, he will probably not want to talk about it much; but that does not mean that he will not appreciate your visit.
Definitely no jollying along! - unless that is how he is approaching it - as I say, take your lead from him, but don't avoid the subject.
As to the little one; does he/she see him at all? If not, unless he is asking to see the child, then it does not really make sense to wheel him/her in to see someone she does not know and then announce that he has gone. And it will depend on the age of the child of course.
I am sure that you will manage to do and say the right things - sometimes in this situation we dread things in advance that actually prove easier than we had feared.