Genially means in a genial way — warmly, with a friendly, happy manner. He smiled genially. She genially offered us tea. It tells you how someone spoke, looked, or acted, not how often they do it. The trap sitting right beside it is generally, which means “usually” or “in most cases.” They are not related in meaning, and mixing them up changes the whole sentence: “He generally smiled” is about habit; “He smiled genially” is about a warm smile this time. Like genial and geniality, this adverb is a little more at home in writing and polite storytelling than in quick everyday chat, where warmly or kindly often feels more natural.