Yes, a Pope can err concerning the faith. Papal infallibility does not mean that every last word of a pope is infallible. I will here adumbrate key points and then leave you with a 14 minute podcast on this topic, which you can take with you as you jog or drive home.
First, distinguish words of a pope from papal words. Words of a pope are, e.g., “Pass the salt” or “I hope Poland wins the world cup.” These are not papal in the least and thus carry no teaching authority whatsoever.
If someone asks the pope his personal opinion on a matter, even a matter of faith, the words the pope speaks are words of a pope, not papal words.
Should we care about words of a pope? Yes. For Almighty God has placed this man in this seat. Thus, we should ignore none of his words. Does caring for his words mean accepting them all as papal words? No, words of a pope are not papal words. That would be a confusion. Does caring for words of a pope mean embracing them all as true? No, it does not nor should not. To do so is the error of Ultramontanism. We must avoid this. It is also called “creeping infallibility”. How should we strike a balance for ourselves? We must always look at the Golden Law of Love. If words of a pope are challenging for us, if say we love our western lifestyle of freedom and spend-the-money vacations, voting pro life, and not thinking about the poor, and if the words of a pope in an interview, say, suggest his opinion that a grave problem today is migration and loneliness, then we should examine our consciences. This man and his personal opinion are perhaps the voice of God calling us to change our western spend-the-money-that’s-mine lifestyle. Of course, prudence should always be at play. But we must let the Holy Spirit challenge us. This means appreciating the prophetic role of the office of pope. However, nor does it mean necessarily endorsing every last personal opinion, even those on the faith. We must be wise stewards of our Tradition. That means first of all and above all knowing our Tradition. The pope and all the bishops are servants of this Tradition. The Tradition norms their decisions and opinions. They do not norm the Tradition.
Now we turn to papal words. Only papal words are authoritative. And only those that are issued under very strict circumstances are infallible. The pope may teach “ex cathedra,” and his words are infallible. He may, gathering an ecumenical council, pronounce with the bishops dogmatically on a point. Then, those words are infallible, the council being approved by him.
Between infallible papal words and non-papal words of a pope are papal words that are not infallible. To call them “fallible” is not a good route to take. For we expect papal words to be true words. However, it may happen that they are at times in error. Really? Hear the podcast on Pope Honorius.
Then should we be really worried? No. The Holy Spirit provides at all times. There have been moments of great and grave confusion in the past. That confusion was certainly not good. No one can make light of it. However, the Spirit soon clarified matters. And further, the faithful always have the Infallible Teachings of Holy Mother Church. Thus, they must always seek out these Necessary Points of Reference. And by these, one will be sure to go right, provided one cleave to God in love and reach out to neighbor in love, not neglecting the poor.