Yes Sally, Pope Francis IS the Pope and is in great need of our prayers!
The Church of Christ is Apostolic and this is also a 'Mark' of the Church.
Specifically it means:
The true Church is also to be recognised from her origin, which can be traced back under the law of grace to the Apostles; for her doctrine is the truth not recently given, nor now first heard of, but delivered of old by the Apostles, and disseminated throughout the entire world. ... That all, therefore, might know which was the Catholic Church, the Fathers, guided by the Spirit of God, added to the Creed the word Apostolic. For the Holy Ghost, who presides over the Church, governs her by no other ministers than those of Apostolic succession. (Tradicat: Marks of the Church Apostolic - Catechism of Trent)The consequence of this is Dogma is that if there are no longer any Bishops, then the promise of Our Lord Jesus Christ that the Church would stand to the end of the world, was false. A secondary consequence of this would be the eradication of the priesthood. Closely related to this Dogma is that defined during the First Vatican Council. Namely, that there will be successors of St. Peter until the end of the world.
The Church in Her wisdom has taken safeguards to ensure that the Apostolic Succession is not lost, specifically by normally having three bishops perform the consecration of a new bishop. Having a consecrator and two co-consecrators ensures that if there is a defect in one, then the other two will supply.
This question is sometimes raised regarding the validity of Archbishop Lefebvre's own episcopal consecration as there were rumours that Cardinal Leinart was a free mason.
The first response is that there were no doubts of the co-consecrators. The second is that the rumours have never been proved. Lastly, the form was followed in the consecration and therefore, barring any authoritative investigation, the principle of the Church is to assume that when the form is followed the sacrament is conveyed.
A similar question is raised concerning Pope St. John XXIII and Pope Paul VI: If they were masons, then they weren't Pope etc.
This question is answered if one adheres to the doctrine (lower case) of the Church on this matter.
Specifically, it is a universally held theological conclusion that if the Bishops of the Church accept a newly elected Bishop as the Pope, then any perceived impediments were not real. This is classified as an infallible dogmatic fact ( Dogmatic Fact or Fancy ).
Caveat, the sede-vacantists that I have encountered throw up lots of excuses to re-imagine this simple doctrine. So if you cast about the internet you will find opinions that run counter to this doctrine. However, I would advise you to take the words of the Church as the Church has understood them and expressed by the theologians of the Church in a consistent and universal manner.