Covered Up: Newman is better known for the songs he's written than the versions he's put out. "You Can Leave Your Hat On" is the most common case.
Misaimed Fandom: Newman is arguably a textbook case: for every fan that gets he's being sarcastic, there are two that don't. Largely it's thanks to his tendencies to compose piano ballads, usually in old-fashioned styles like ragtime, that make his music seem like easy listening. It doesn't help that the irony is sometimes very subtle: "Jolly Coppers On Parade" is a good example.
Recycled Script: Some feel he has a tendency to recycle his songs' themes, though this is largely with his film songs and scores, and less so with his independent albums.