[FEN "1nkr2nr/pppp2pp/2b2p2/3q4/3P4/1NQ5/P1P2PPP/R1B1R1K1 w KQkq - 0 1"]
Can White possibly win in this position and avoid a queen mate on g2?
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Avoid queen mating on g2 is easy:
Winning this position is objectively speaking impossible. Black is up a piece and white does not have sufficient compensation for it. Most players would resign if playing somebody of 1800 strength or even less.
If black is a beginner and/or short on time white could try to continue the game. The usual plan in lost positions with material down is to keep the position as complicated as possible and to avoid trades. You want to activate your pieces and start to attack something, specifically the enemy king.
In this regard one possible plan could be:
Once you get in c4 you can follow up with d5 which limits the range of the only active black minor piece.
Of course black also makes moves in-between and could spoil this plan.
The following example continuation is not best play, but about the best white could hope for for the next couple of moves:
[FEN "1nkr2nr/ppp3pp/2b2p2/3q4/3P4/1NQ5/P1P2PPP/R1B1R1K1 w - - 0 1"] 1. Qg3 g6 2. Bf4 Na6 3. Rac1 Kb8 4. c4 Qf7 5. d5 Ba4 6. Nd4
The final position is still lost for white, but at least white got the more active pieces and more space than black. There could be some attacking potential along the b file as well, now that the bishop left c6.