does sprite queen spawn a sprite every turn?

Yes — while Sprite Queen stays on the board, she makes a ready 3-Might Sprite token with Temporary at the start of each of your Beginning Phases (plus once more when you play her).

The old Sprite from last turn is killed by Temporary at that same moment, before scoring, so you effectively get a fresh Sprite every turn. Note the trigger only fires at your Beginning Phase and only if Sprite Queen is still in play when it starts — if she's killed before then, no Sprite that turn. Since you control both triggers (the old token's Temporary death and her spawn), you choose their order — put the spawn on the Chain first so the death resolves before the new Sprite enters.