A queen against a pawn one step from promotion is usually winning, but only through the right technique: keep checking until the enemy king is forced in front of its pawn, then use that tempo to bring your king closer. Against a bishop or rook pawn this trick fails, so knowing when it works is essential.
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You have a queen; Black has a king and a pawn on e2, one step from queening. Plenty of games have been drawn here by a player who could not stop the pawn in time. Your king stands far away on b5, the queen on h4. The win comes in cycles: check, gain a tempo, bring the king closer, repeat.
What you will learn
01Why a check that drives the black king to e1, in front of its own pawn, is worth a free move.
02How to use that free move: the king walks while the queen keeps guard over e1.
03Which pawns beat this method and why: the stalemate corners of a bishop or rook pawn.
The technique
The pawn cannot promote while the queen watches e1, and Black's king has to protect the pawn and keep e1 free at the same time, which is one square too many. Check from squares that leave the king nothing but e1. When it steps there it blocks its own pawn, and you spend the move on your king: b5 towards c4, later d3. Two or three cycles and your king stands beside the pawn; the pawn falls or a check becomes mate.
Where it goes wrong: with a bishop pawn (c- or f-file) or a rook pawn (a- or h-file) the black king has a corner to hide in, and taking the pawn, or boxing the king, is stalemate. The queen alone then usually draws unless your king is already close. Here the pawn is on the e-file, so the method works.
How to practise it
You are White. Before each queen move ask one question: after this check, does the black king have a square other than e1? If yes, find a better check. Move the king, not the queen, whenever the pawn is blocked. Winning the pawn inside eighteen moves ends the drill.
No. Against a knight pawn or a central pawn it is, almost regardless of where the attacking king starts. Against a bishop or rook pawn it is usually drawn by stalemate unless the attacking king is close.
Why not just take the pawn?
Because the black king on d2 protects it. The queen wins the pawn only once your king arrives, or with a check that also attacks the pawn.