Can Black somehow get a stalemate against the White King-Queen combination or should he try to save his pawn as long as he can?
8/8/5K2/4Q3/4p3/3k4/8/8 b - - 0 1
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No way. I would resign if I were Black. This is a hopeless position - game over. Anybody other than a total beginner would win for White.
It'd be possible to draw by stalemate.
Note: colour reversed in the examples.
King + rook pawn vs. queen
7K/7P/6q1/8/8/1k6/8/8 w - - 0 1
King + bishop pawn vs. queen (variation)
7K/5P2/6q1/8/8/3k4/8/8 w - - 0 1
Black's only chance to salvage a draw is to of course, keep his pawn with the hope of queening it. However, this is not possible here because after, say, 1...e3 2.Qd5+ Kc2 3. Qe4 Kd2 4.Qd4+, black must block his own pawn with 4...Ke2 and white can play 5.Kf5, bringing his king closer to the pawn. After 5...Kf2 for example, white goes 6.Qf4+ and black needs to go back with 6...Ke2, and white brings his king closer again with 7.Ke4 and plays 8.Qxe3 the next move.
This idea is similar with the pawn on the 2nd or 7th rank (i.e. 1 square away from queening), except for rook or bishop pawns, which can be drawn under certain conditions (the side with the queen needs to have his king in what most endgame textbooks call the "winning zone"). Wikipedia actually has an article on this.
Without going into a lot of detailed analysis, the simplest way to win as White is to get the Q in front of the pawn (1. Qa1 followed by 2. Qe1 will do it) and then bring your K up to win it with the Q's help. After that it's a simple win with K and Q vs K. Forget about any stalemate ideas. They're wishful thinking in this position.
Black gets a stalemate if his pawn is a rook or bishop pawn, it's on the 7th rank, and the white king is not inside the winning zone (usually one or two squares from the pawn - see Jesus De La Villa's 100 Endgames You Must Know for more details.)