The euphemism is "love child". I wonder how loved the children of recent celebrities feel? Would "power child" work better?
I have a fantasy that the next man who drags his dirty laundry across our media pages will say this:
"Today I am stepping out of the spotlight. I am asking my wife for a divorce so that I can reorganize my life and offer the love and support that all of my children deserve. I will embrace my children, both those born within my marriage and those born outside my marriage with the same affection and provide them the same support. My home will be their home when they visit me. My life will be connected to theirs. I will be a father to them all. I realize now that sexual intercourse is a commitment, and a consequence of that commitment is an obligation to children formed in that act. I do not have a moral choice to claim that my behavior was a mistake that I regret. It would be wrong to tell any child that he is the outcome of a mistake."
Its unlikely to happen that way.