Wait--I didn't look close enough. This line:
$cart->writeRow($currentRow, 0, $item);
should be:
$cart->writeRow($currentRow, 0, $array_to_write);
Edit: Like I said, that should work, but you could also condense the whole thing to this:
$currentRow = 4;
while ($row = mysql_fetch_assoc($result))
{
$array_to_write = array($row['client'], $row['director'], $row['manager']);
$cart->writeRow($currentRow++, 0, $array_to_write);
}