Covetously is the adverb: you look, stare, speak, or act in a covetous way — with a strong wish to have what someone else has. It tells you how the action is done. Someone may gaze covetously at a shop window, or talk covetously of a brother's success. It is not a common everyday word; it sounds bookish, and many speakers would just say longingly, greedily, or with envy. Though actually, greedily can miss the target: greedily is about taking a lot, while covetously keeps the idea that the thing already belongs to someone else, or is out of reach and wanted as if it were.