Correct spelling, explanation: word publicly is an adverb created by adding suffix -ly to an adjective public. This case is an exception because the general rule in creating adverbs from adjectives with ending ic is that we add a suffix -ically. In that case, the version publically could be correct, but we should remember that the correct one is publicly.

Definition of publicly:
adverb: something that is done openly, so everyone can hear and know about it
He was not afraid to speak out publicly about the fraud scandal in his company.
– something that is paid, done or owned by the government
The new road in our town will be publicly funded.

Incorrect spelling, explanation: the general rule in creating an adverb from adjectives with ending ic is that we add the suffix -ically. If that rule were applicable, the word publically would be correct. However, the adverb publicly is an exception and we create it by adding the suffix -ly to the adjective public. Therefore, the correct version is not publically but publicly.