Kinda hard.
The Pope is the Bishop of Rome after all.
However, I'll try to avoid the Schism or delay it enough in order to let the idea of Avignon as the proper and permanent see for the Papacy getting firmly established.
There was quite a lot of unhappiness about the whole "Pope in Avignon" thing. It is called "Captivity" for a reason, although it's probably not a completely fair label.
Time would smooth things down, maybe, in some centuries.
Effects are quite huge. The whole dynamic of international relationships between the Pope and the European monarchs is deeply changed. Reformation as we know it is butterflied away, of course, but any analog movement will be bound to be deeply affected by the fact that the Pope is quite in, er, France (technically Avignon was not part of France at this point, but still). And probably he is no longer the independent ruler of an independent Italian power on top of his spiritual authority.
Probably this would mean a lot less of said authority being recognized, as in the Pope will be seen quite as a "French" political tool. His relationships with Paris might not be rosy all time, but no Avignonese Pope is likely to have the independent leverage their predecessors in Rome enjoyed face the Holy Roman Emperors.