I read a book back in the 90's most likely from a school library. The main character was some lost prince who travels to an ice planet for some reason. The cover had him on the ice planet with some alien rifle. I believe he may have looked like an elf with his facial features.
Perhaps WINTER WORLD, by C.J. Mills, a 1988 paperback? It's been thirty years or so since I read it and I can't find my copy to quote from, but the cover seems a reasonable match, and I do recall it involving a member of a royal family returning to the ice planet of his family (I think he'd been away at school or in exile or some such) and sorting out some sort of challenge to their rule.
Mills is/was a local (Twin Cities) writer, which is the only reason I read the book in the first place; the only other plot point I recall is the protagonist, being chased by enemies, "inventing" the skill of skiing to escape them -- which struck me as, shall we say, pretty unlikely for a space-faring society whose home base was an ice planet.