Correct spelling, explanation: adverb publicly is created from an adjective public by adding a suffix -ly. However, it is an exception. The general rule for adverbs created from adjectives ending with ic is to add the suffix -ically. In that case, the version publically could be correct if it wasn’t an exception. The only correct form is publicly and not publically.

Definition of publicly:
adverb: something done in an open manner, that everyone can know and hear about it
The company publicly apologized for the breach of data and presented a new privacy policy.
– something owned, done, or paid by the government
The new part of the airport will not be publicly funded.

Incorrect spelling, explanation: publically is an incorrect version of the adverb publicly. Although the general rule for adverbs created from adjectives with ending ic is to add a suffix -ically, in this case, is it not applicable. Publicly is an exception and is created by adding a suffix ly to an adjective public. For that reason, the correct spelling is publicly and not publically.