Which city became the center of the catholic church in the early 1300s?

Avignon

At the time, Avignon was in the Kingdom of Arles, which was part of the loosely-confederated Holy Roman Empire, bordering on France.  (Today it is within the borders of France.)

There had been struggle between Pope Boniface VIII and the French king Philip IV over control of the church in France.  Philip actually sent men to rough up Boniface during that time.  After Boniface's death and then a papacy of less than a year by Benedict XI, pressure from France resulted in the electing of a French cardinal as Pope Clement V, in 1305.  Clement moved the office of the papacy from Rome to Avignon, where it stayed from 1309 to 1376, with seven popes total governing the church from there.