What does "cavallier extension claim" in The Algorithm Design Manual mean
The phrase means "cavalier" in the ordinary English sense of the word, i.e., it is talking about a claim in a proof that is made carelessly.  Without more context or examples I'm not sure exactly what pattern of flawed proof it is referring to, but it has something to do with making an assumption/claim (without justifying it) that the solution for $n+1$ is related to the solution for $n$ in a particular way.

"Cavallier extension claim" is not a technical phrase.  It's just some English that the author is using.