Puzzle 1 from a game Li Laiqun - Hu Ronghua

The correct move is 1. ... C5.9

This looks like a simple endgame. Black has a cannon more
and should win. In a first step he wants to win Red's pawn or
exchange a cannon pair. However, Black has to be aware,
that a cannon can't win against the naked king without
the help of an advisor. King, cannon and advisor against
naked king is an easy win (actually, they can even win
against king and two advisors), but against king, cannon and
elephant (even two elephants) the naked king can hold a draw!

In the game Black played the careless 1. ... C1.4? and after
2. P6+1 he realised that he couldn't win the game anymore.
In both of the lines
2. ... C4+7 3.P6.5 C4-7 4.P5+1 (Answer1)
and
2. ... C5+1 3.C6+7 A5-4 4.P6+1 C5.9 5.P6+1 (Answer2)
Red has eliminated both of Black's advisors and the remaining
cannon can't win any more.
2. ... C5+1 3.C6+7 A5+4 is hopeless as well, e.g. 4.C6.8 C5-2
5.C8-7 K6.5 6.C8.6 (Answer3)

1. ... C5.9 or many other moves of this cannon would have allowed
an easy win. Now, Red has to face the threat 2. ... C1.4 and
has no defence. If Red's cannon moves sidewards, e.g.
2.C6.1, Black will win the pawn after 2. ... C1.4 (Answer4).
Going forward with the cannon isn't better, 2. ... C1.4
will still win the pawn or the cannon (Answer5).
1. ... C5.9 2.P6+1 A5+4 is hopeless as well (Answer6)
and 1. ... C5.9 2.K6.5 C1.5 neither helps.
     

Puzzle 2 from a game Huang Yong - Yan Mujiang

The correct move is 1.R6.3

Both kings are in vulnerable positions. Furthermore, Red's rook
on the fifth file is under attack from Black's cannon. Red
played 1.R9.6? hoping for 1. ... C4-6 2.R6+1 K6+1 3.R6.4 K6.5
4.R4-8 with a good position for Red (Answer1). However, Black
played 1. ... R2-8 leaving Red's rooks in despair. 2. A5-6
is still not possible because of 2. ... R2+7 checkmate. Therefore
Black will win at least an exchange (rook for cannon). Red
tried 2.R++1 K6+1 3.R+-1 K6+1 4.P3+1 and resigned after
4. ... C4-6 5.P3+1 R2+7 6.K6-1 R2+1 7.K6-1 R2.5 (Answer2)

With 1.R6.3 Red could have reached an advantageous position.
See 1. ... C4.7 (or 1. ... E7+5 2.R3+2 C4.7 3.R9.6 with similar
lines, 2. ... R2.5 is too slow because of 3.R9.4 R6-7 4.R3+1 mate)
2.R9.6 (one move before this idea was insufficient, now it's
good!) 2. ... C7-6 3.R6+1 K6+1 4.R6.4 K6.5 5.R4-8 (Answer3) and 
Red is better, although he needs some moves to consolidate his
king's position.
     

Puzzle 3 from a game Liu Dahua - Li Laiqun

The correct move is 1.C5.6

With 1.C5.6 Red could have started a decisive attack against
Black's king, see e.g.
1.C5.6 K4.5 2.H6+4 R3-3 3.H4+3 K5.4 4.R8.6 H2-4 5.R6+2 A5+4 
6.R6+3 checkmate (Answer1)
1.C5.6 R3.4 2.K5+1 K4.5 (or 2. ... A5+4 3.H6+7 C9.3 4.R8.7 A6+5 
5.H7-8, Answer2) 3.H6+4 C9.6 4.R8.7 R4-5 5.R7+7 R4.3 6.H4+6 checkmate
(Answer3)

Instead, Red played 1.C5.8?, probably with the plan to exchange
the cannon against the horse and to win Black's promoted pawn.
However, this plan didn't work out:
1.C5.8? R3.4 2.H6+7 K4.5 3.C8+2 R4+1 (Black's promoted pawn
will survive!) 4.R8.5 P2+1 (Answer4) Black has strong advantage and
won the game later on.