Correct spelling, explanation: heavily is the correct form, as it is an adverb made of an adjective heavy. In order to create an adverb from an adjective we need to add the suffix -ly, and in the case of y at the end, it is changed to i – this exact thing happens in heavily. Heaviliy is thus a mistake, probably a typo.

Definition of heavily:
adverb, thickly, tediously
He was breathing heavily after running to the bus.
My mum dresses salads too heavily for me.

Incorrect spelling, explanation: there is no need to add extra i in this adverb. To create an adverb from the adjective heavy we add -ly suffix to the word. Heavy ends with y, so we also have to change this letter to i, so the result is heavily. Heaviliy is a mistaken form, as the suffix is -ly, not -liy. Therefore heavily is the correct spelling.