Push the pawns, keep your king and queen together and try not to get into a perpetual. You probably can give up the backward pawn, it's not really helping anything. Sure it requires some understanding of board geometry but it's nothing out of this world. There's no particular theory, just experience and some calculation. You don't have to play tablebase/engine-perfect moves.
Against the machine this can be very tricky of course, but a human will likely walk into a fork sooner or later.
https://en.wikipedia.org/wiki/Queen_and_pawn_versus_queen_endgame
This might help
Hello!
I solved this puzzle today. The puzzle ended after the 4th move or so.
Even though I could intuitively tell that the position was winning, I had no idea how to actually win the rest of the game, so I tried to play the position against the engine and immediately blundered and lost my pawn every single time. I then spent another hour analysing each move with the help of the engine, and I finally got a solution on how to win the rest of the game with an opponent who plays optimally until White is forced to trade Queens.
To a 1200 player like myself, the calculations were mind-blowing. I loved how both sides used repeated checks to reposition their Queens, each anticipating the other one's threats.
My question is, is there any theory on how to play such Endgames, like movement patterns or something, or is it just about being good at calculations? At what level is one expected to learn such things?