The Pope as the Guardian of Revealed Truth, Not Its Arbiter
Doesn’t the Pope have the authority to allow the divorced and “remarried” to receive Holy Communion?
No. Since the Church’s teachings come from God Himself, not even a Pope has the authority to change them. Nor does he have the authority to relax any of God’s commands requiring us to order our lives upon His truth. The Pope does have the authority to establish or change Church discipline in some matters, but even then only in ways consistent with the revealed Truth and the salvation of souls.
The provision of Canon Law forbidding persons who obstinately persevere in manifest grave sin to be admitted to Holy Communion, is derived from the divine law. Therefore not even a Pope can introduce legislative changes or pastoral practices which would oppose this fundamental law and teaching of the Church.
Are any Church leaders offering sound guidance to the faithful in these matters?
Yes. Cardinals Carlo Caffarra and Raymond Burke, and Bishop Athanasius Schneider are among the prominent signatories of the Declaration of Fidelity to the Church’s Unchanging Teaching, and to Her Uninterrupted Discipline, on Marriage. The hundreds of faithful clergymen who have signed this declaration affirm that while the plagues of divorce and sexual depravity spread everywhere, even within the life of the Church, it is the duty of bishops, priests and Catholic faithful to declare, with one voice, their fidelity to the Church’s unchangeable teachings on marriage, as well as to Her uninterrupted discipline, as received from the Apostles — namely:
That a well-formed conscience will never reach the conclusion that, given a person’s limitations, his remaining in an objectively sinful situation can be his best response to the Gospel, nor that this is what God Himself is asking from him;
That people cannot look at the Sixth Commandment and the indissolubility of marriage as mere ideals to strive after;
That personal and pastoral discernment can never lead divorced persons who are civilly “remarried” to conclude that their adulterous union can be morally justified by “fidelity” to their new partner, that withdrawing from the adulterous union is impossible, or that, by doing so, they would expose themselves to new sins;
That divorced persons who are civilly “remarried” and who cannot satisfy the grave obligation to separate, are morally obliged to live as brother and sister and to be particularly vigilant not to give scandal;
That there is no halfway point between being in the grace of God and being deprived of it by mortal sin, and therefore that spiritual growth for someone living in an objective state of sin consists in abandoning that situation;
That the complexity of situations and the varying degrees of responsibility among cases do not prevent pastors from concluding that those in irregular unions are in an objective state of manifest grave sin;
And that divorced persons who are civilly “remarried” and remain in their objective state of adultery, can never be considered by confessors as living in an objective state of grace and entitled to receive absolution or to be admitted to Holy Communion, unless they manifest true contrition by a firm resolve to abandon their state of life. 
Many lay theologians, too, such as Prof. Peter Kwasniewski, have come forward to assert the plain truth about today’s unprecedented situation — namely, “To the extent that the Buenos Aires guidelines [for the implementation of Amoris Laetitia] permit what the Church has always forbidden as a matter not of human law but of divine law, they are false, and any true Catholic will reject them without hesitation.” 
Seeking the Peace of Christ
Could it ever be permissible for a validly married divorced Catholic who has entered into a new intimate relationship (or even a civil “marriage”) to continue cohabiting with that new partner?
For such persons, the duty to cease giving public scandal, to separate from the illicit partner, and to return to one’s true spouse (when possible) is of the first importance. But in the case of a person whose true spouse does not desire a reunion, and who has had children with the new partner, there may be some freedom in this regard. Pope John Paul II describes such a possibility for “those who, repenting of having broken the sign of the Covenant and of fidelity to Christ, are sincerely ready to undertake a way of life that is no longer in contradiction to the indissolubility of marriage. This means, in practice, that when, for serious reasons, such as for example the children’s upbringing, a man and a woman cannot satisfy the obligation to separate, they take on themselves the duty to live in complete continence, that is, by abstinence from the acts proper to married couples.” 
Of course, their obligation to avoid scandal in such a case would be all the more serious. Not only must they refrain from any display of intimacy proper to married couples, but (since their condition as divorced and remarried may be widely known in the community) they should also take care to make it widely known that they no longer live as husband and wife, and instead live chastely together, under the same roof but not in the same bed.
How can divorced Catholics offset the many dangers and temptations that assail them?
One means of doing so is to concentrate on the deeply rewarding work of raising their children properly. Another important way is to build up an especially fervent spiritual life. For this latter, a spiritual director’s advice will be helpful:
“Idleness, as the inspired Scriptures say, is a danger to anyone, but it is particularly such to divorced persons. By idleness is not meant only sitting around doing nothing, but also going around doing useless or unsuitable things. A divorced person who spends a great deal of time at mixed social gatherings, is not only wasting time, but creating occasions for temptations.
“No matter how young a divorced man or woman may be, no matter how bleak and difficult the long stretch of the future may seem under the restrictions to which a divorced person is subject, God’s grace is ready in abundance to make virtuous living possible and even easy for such a one. However, this grace must be sought, and sought with a zeal proportionate to the soul’s need.
“Therefore divorced persons should go to Confession at least every few weeks in order to keep making a check on whether they are carrying out God’s strict commands. They should receive Communion often, even every day if possible, knowing that this Sacrament will be the means through which they can grow enough in the love of God to resist all temptations to sinful love. And they must adopt a firm schedule of daily prayer and spiritual reading so that they may develop a genuine interior life — that is, one in which they grow constantly in conscious union with God.” 
 Cf. http://www.filialappeal.org/ “Why it’s impossible for the Catholic Church to ever accept remarriage,” April 17, 2018, LifeSiteNews; https://www.lifesitenews.com/blogs/why-its-impossible-for-the-catholic-church-to-ever-accept-remarriage  Familiaris Consortio, 84  Rev. Donald Miller, C.Ss.R., Program for Divorced Catholics, Liguorian Press, 1962.