How will the fall of rome affect the papacy?

Do you mean how did the fall of Rome affect the papacy? There was conflict with Theodoric the Great who created a Ostrogoth kingdom in Italy and in today's former Yugoslavia. However, this was not due politics. Theodoric wanted to preserve the Roman Empire and did not interfere with Rome or the Latins of Italy. It was about religion as Theodoric refused to convert to Catholicism and defended the Aryan creed of his people. There was also conflict with the Romans of the east, again, over religion. There was tension between Catholicism and Orthodox Christianity.

Later there was conflict with the Lombards who invaded Italy. The pope got Charlemagne, the king of the Franks, to get rid of them. In return he gave him Italy and crowned him emperor. This was the beginning of the Carolingian Empire and of the concept of the temporal power of the pope. Charlemagne also gave the pope land in Italy which became the nucleus of what was to became the Papal States.