The Hindu is sort-of centre-left

In Malayalam(main language of Kerala), there’s:
https://www.deshabhimani.com/
http://www.janayugomonline.com/

They were founded by leftist parties, so there might be not enough critcism of the left. But exposure to regular media will balance that more than enough.

I think there’s similar ones called Theekkathir, People’s democracy etc.