1. When is the right time in a relationship, for a prospective partner to tell the other that he or she has a child?
2. How should he or she go about giving out the information?
Barring in mind that in a christian faith relationship, nothing should be hidden. But revealed with timing and wisdom.
I don't care for going on dates to begin with (just my view for myself), I prefer to start as friends and let that turn into romantic feelings. Or not, depending.
I break the number one rule of dating. I don't hold back about all the struggles and problems I contend with (and it's a lot).
Of course I think how you present it can have an affect. If you go on and on, especially in an emotional state, that could be a turn off.
However, if you state things matter-of-factly and dont dwell too long, unless asked questions.
You may not want to dump it all at once, but prioritize and share over a few dates. Having a kid should be top priority on your list. Some don't want children, others don't want to be a step father. But I would not introduce them to your child early on. Let the relationship grow and stabilize. It's easier on children when things go south in a relationship if they don't know the guy. So introducing them early is a bad idea.
So far I cannot recall anyone who has loss interest after I shared with them. It could happen, but it hasn't happened to me, with some even talking marriage.
Full honesty, no holding back, is my view. If they lose interest, good, you've weeded out someone that would only be resentful if they stayed, or just leave you after you're already invested in that person.