We use both right and rightly as adverbs, but they are used in different ways and with different meanings.
Right is an adverb of manner. It means ‘correctly’ or ‘well’:
The calculation was wrong first time, but I did it right the second time.
Not: … I did it rightly …
When everything goes right, no one says thank you, but when things go wrong, they start complaining immediately.
We use rightly to give an opinion or viewpoint on something. It usually comes in the normal mid position for adverbs (between the subject and the main verb, or after the modal verb or first auxiliary verb, or after be as a main verb). It means ‘in the opinion of the speaker, in a morally correct way’, and is more subjective than right:
He rightly took the money he had found straight to the nearest police station. (In the speaker’s opinion, this was the morally correct thing to do.)
Climate change is rightly seen as the greatest threat to the world at the present time.
