Okay, I don't know what exactly you're struggling with so this might not be helpful.
The issue that this is dealing with is where you have a situation where someone goes, for example (from the case we read about this), "I'll give you 10K if you promise not to file that paternity suit against me." What happens if that paternity suit could not possibly have prevailed -- the guy's not the father of the baby? Is not filing a lawsuit that would have been groundless to begin with consideration?
So the first part is saying that there needed to have been some actual doubt about the validity claim when the contract was created, OR the person who's saying they're not going to file the claim at least needs to believe there is -- basically a good faith requirement. The second part I THINK is dealing with clauses in settlement agreements where the person isn't even necessarily contemplating filing such a claim (they don't necessarily think that there's a valid claim there), and in that case the writing serves as consideration. Make sense?