Pope Francis would seem to be referring to cases where there is grave matter, but not the other two criteria for mortal sin
In the text preceding this note, Pope Francis observes that, while certain individuals may be objectively in sin, they may not be fully culpable. This is nothing new; the Church has long taught that mortal sin requires the presence of three criteria: grave matter, full knowledge and freedom of the will (CCC 1857). So the pope is saying that, though grave matter is always present in an irregular union, the other two criteria may not be.
In such cases, the pope says, the Church can not merely state a rule as though it were “a stone to throw.” Rather, it must be a source of help for the couple to “grow in the life of grace.” And then he adds this footnote:
In certain cases [emphasis added], this can include the help of the sacraments. Hence, “I want to remind priests that the confessional must not be a torture chamber, but rather an encounter with the Lord’s mercy.” … I would also point out that the Eucharist “is not a prize for the perfect, but a powerful medicine and nourishment for the weak.”
In “certain cases,” but which? If the text that precedes the note is of any help, the pope would seem to be referring to cases where there is grave matter but not the other two criteria for mortal sin. If there is no mortal sin, nothing bars one from the Eucharist. Only a pastor who knows and has counseled the individuals in question can make this determination.
Canon lawyer Edward N. Peters, in an article that’s worth your time, points out, however, that the pope cannot, in an apostolic exhortation, change the law about withholding the Eucharist from those who (as canon 915 puts it) “obstinately persevere in manifest grave sin.”
Still, it must be said that the language surrounding footnote 351 does not at all describe people who “obstinately persevere in manifest grave sin.” The pope speaks of those who are not fully culpable due to the presence of “mitigating factors.”
Nor does the pope say that priests should allow couples to remain in sin once they are convicted of the grave matter. He speaks elsewhere (cf. §222) of the need for pastors to help couples develop a “fully formed conscience.” One the conscience is formed, the grave matter must end.
And here is where footnote 329 becomes of help. In this section, the pope restates a point that St. John Paul II had made in Familiaris Consortio 84, which is that the good of children might mean that couples in an irregular union cannot separate (i.e., divorce or live apart). John Paul II is at pains to point out that celibacy is required of couples in this situation.
In note 329, Pope Francis adds:
In such situations, many people, knowing and accepting the possibility of living “as brothers and sisters,” which the Church offers them, [emphasis added] point out that if certain expressions of intimacy are lacking, “it often happens that faithfulness is endangered and the good of the children suffers.”