Build a bit mask for the byte that contains the position. Then count the bits in the bytes before the position, and do a bitwise and for the byte that contains the position.

        my $mask = pack "V", 2 ** ($p % 8) - 1;
        my $c = unpack( '%32b*', substr( $v, 0, $p / 8 ) )
              + unpack( '%32b*', substr( $v, $p / 8, 1 ) & $mask );
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(Update: removed a superfluous pair of parens in the last line.)