user29730: Consider this chess board set up:

Is the black king allowed to move to B4 or D4? The knight on C6 would put their white king in check if it moves.
I haven't really read any rules about this so thought I'd ask.

user8888: You are in checkmate if you are in check and have no legal move by which to escape it.
In this case you cannot escape check by moving your king. B4 or D4 would leave you in check from the knight, C4 would leave you in check from the bishop and anything on rows 2 or 3 would leave you in check from a rook.
However moving your king is not the only way to escape a check. It is also sometimes possible to escape a check by blocking the capture route or by capturing the attackers piece. In the original version of the question it was possible to block the capture route by moving your queen to D3. So the position was not checkmate. In the revised version of the question it is not possible to block the attack path or capture the attacker so the revised position is checkmate.

user12523: Yes, a piece can check even if it can't move because of a pin. This is explicitly written in the FIDE Laws of Chess (emphasis added):

3.9.1 The king is said to be 'in check' if it is attacked by one or more of the opponent's pieces, even if such pieces are constrained from moving to the square occupied by the king because they would then leave or place their own king in check.

Therefore, if the black king tried to escape to b4 or d4 it would be moving into check, which is not allowed.

29730: <p>Consider this chess board set up:</p>
<p><a href="https://i.stack.imgur.com/GkpUh.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/GkpUh.png" alt="enter image description here" /></a></p>
<p>Is the black king allowed to move to B4 or D4? The knight on C6 would put their white king in check if it moves.</p>
<p>I haven't really read any rules about this so thought I'd ask.</p>
 8888: <p>You are in checkmate if you are in check and have no legal move by which to escape it.</p>
<p>In this case you cannot escape check by moving your king. B4 or D4 would leave you in check from the knight, C4 would leave you in check from the bishop and anything on rows 2 or 3 would leave you in check from a rook.</p>
<p>However moving your king is not the only way to escape a check. It is also sometimes possible to escape a check by blocking the capture route or by capturing the attackers piece. In the original version of the question it was possible to block the capture route by moving your queen to D3. So the position was not checkmate. In the revised version of the question it is not possible to block the attack path or capture the attacker so the revised position is checkmate.</p>
 12523: <p>Yes, a piece can check even if it can't move because of a pin. This is explicitly written in the <a href="https://handbook.fide.com/chapter/E012018" rel="nofollow noreferrer">FIDE Laws of Chess</a> (emphasis added):</p>
<blockquote>
<p>3.9.1 The king is said to be 'in check' if it is attacked by one or more of the opponent's pieces, <strong>even if such pieces are constrained from moving to the square occupied by the king because they would then leave or place their own king in check</strong>.</p>
</blockquote>
<p>Therefore, if the black king tried to escape to b4 or d4 it would be moving into check, which is not allowed.</p>
