When we talk about the phrase "gnaw away," the first synonyms which come to the mind are corrode, erode, consume, grind down, wear down, chip away, and bite at. All these words have a common meaning of damage or eat away at something slowly and continuously. Another similar phrase which could be used instead of "gnaw away" is "chisel away," meaning to slowly carve out or chip away at a piece of material. Other synonyms for "gnaw away" could be nibble, chew, peck, pick, and scrape, which all indicate a similar action of gradually eating away at something.