After checking a king

hi Mike! Thanks for the quick response. Is it my responsibility to check them again though? I was told I need to notify them that they moved into another checked spot so they can make another move.

In tournament chess you never talk to your opponent (unless it's to offer a draw) so you don't warn your opponent. If they make an illegal move then you stop the clock and make a rule violation claim with the tournament director.
Illegal moves in tournament blitz games often count as an immediate loss.
Illegal moves in long tournament games usually result in a small time penalty.
So if you're asking about the rules of chess, that's how it works.
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But other than literal rules, there are also unspoken rules of etiquette for casual games.
In casual games between inexperienced players, it's often polite to announce "check" to let them know the king is under attack, but experienced players don't do this even in casual games.
In casual games between any level of players, it's often polite to allow your opponent to take back illegal moves, but if the two players are very competitive with each other, then they may decide to handle illegal moves as if they were playing in a tournament. Since treating a casual game so strictly is uncommon, it's polite to agree to tournament rules before the game begins so that both players understand.
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To reiterate, if I were playing someone in a casual game, I would not announce check, but if they made an illegal move, I would point it out and let them play something else without any kind of penalty.
If I wanted illegal moves to count as a loss, I would let them know before the game began.
FIDE Laws of Chess:
"7.4 a. If during a game it is found that an illegal move, including failing to meet the
requirements of the promotion of a pawn or capturing the opponent’s king, has
been completed, the position immediately before the irregularity shall be
reinstated. If the position immediately before the irregularity cannot be determined
the game shall continue from the last identifiable position prior to the irregularity.
The clocks shall be adjusted according to Article 6.13. The Articles 4.3 and 4.6 apply
to the move replacing the illegal move. The game shall then continue from this reinstated position.
b. After the action taken under Article 7.4.a, for the first two illegal moves by a player
the arbiter shall give two minutes extra time to his opponent in each instance; for a
third illegal move by the same player, the arbiter shall declare the game lost by this
player. However, the game is drawn if the position is such that the opponent cannot
checkmate the player’s king by any possible series of legal moves."
https://www.fide.com/FIDE/handbook/LawsOfChess.pdf
Hi there,
URGENT! After checking a king, if the other player moves their king to another checked spot, do i have to warn them check again? Or can i go ahead and kill the king?
Thanks for your responses!!