ZTG, a link to your website would be very useful. It could be that something is wrong and that PHP is suppressing the errors for one, or that the captcha image is not being displayed because it's doing some weird stuff.
dymofff, no, the absolute path meaning something like the correct path to the file. If, when you write require_once('./captcha/php-captcha.inc.php');, it has an error message that says "failed to open stream: No such file or directory" it means exactly that - you're telling PHP to look for a file that doesn't exist.
EDIT: Actually, based on your error message you had posted at the beginning, we can tell you that the following code will work in any file:
However - this is important! - don't add this code to show_news.php if you're displaying CuteNews in another file with something like
The code above needs to be added at the very beginning of that file. Otherwise you will get the error you had in post #5 again. This appears to be a file called blog.php on your website. The code needs to go in there.