Is there any method to determine if the weaker side in KQPKQ has a perpetual check? Is there any possibility to check, without the use of the tablebases, if a position in the endgame queen+pawn vs queen (KQPKQ) allows a perpetual check for the weaker side?
Whether there is a draw in hand depends on both the rank and file of the pawn in the following analysis.
The following as an analysis quoted in Wikipedia's article, queen and pawn versus queen endgame. All quotes and diagrams are taken from Wikipedia. Wikipedia cites table base analysis by the chess computer Belle from 1985, alongside John Nunn, Reuben Fine, Pal Benko and others.
Key For Diagrams

Black king on white dots: win with White to move, draw with Black to move; on black dots – win with either to move; other squares – draw

Rook Pawn

A rook pawn needs to be on at least the sixth rank to have decent winning chances


Knight Pawn

A knight pawn should be on at least the fifth rank to have good winning chances. A knight pawn on the fifth rank has better winning chances than a rook pawn on the sixth rank. There are two reasons for this:

The king has squares on the adjacent rook file to try to avoid perpetual check
The exchange of queens is less likely to lead to a drawn king and pawn versus king endgame



Bishop Pawn

A bishop pawn offers the best winning chances. One reason is that there is no drawing zone in the opposite corner for the black king if the pawn is on at least the fourth rank. If the pawn is on the fifth rank the defender's chances are small unless the king is in front of the pawn. A pawn on the sixth rank wins unless the defending king is in front of the pawn


Centre Pawn

A central pawn has better chances to win than a rook pawn or knight pawn, but not as good as a bishop pawn. As with the bishop pawn, there is no drawing zone for the defending king in the opposite corner. It is better for the defending king to be on the &quot;short side&quot; of the pawn rather than the &quot;long side&quot;