Chouby on "[Plugin: Polylang] Translations on homepage"

It's hard to know why you got this error (and not me...). But I updated the file again, corrected some bugs and it was since successfully tested by another person than me. So you can test it again if you want.

Otherwise, you can try to replace your code by:

<?php if (is_home())
{
	$slug = 'puzzlenews';
	if (($cat = get_category_by_slug($slug)) && ($tr_id = pll_get_term($cat->term_id)) && !is_wp_error($tr = get_category($tr_id)))
		$slug = $tr->slug;
	query_posts("category_name=$slug");
}
?>

In that case, the best is to make a child theme.