Amoris Laetitia nn.304-306

Because of forms of conditioning and mitigating factors, it is possible that in an objective situation of sin - which may not be subjectively culpable, or fully such - a person can be living in God's grace, can love and can also grow in the life of grace and charity, while receiving the Church's help to this end. [AL n.305]

In context, this refers to "irregular" marital situations such as those of the person who has divorced and re-married. In this situation, the Church's discipline does not allow the person to receive Holy Communion.

But I do think we need to be careful in how we understand the grounds for the Church's discipline in this regard. It represents a particular application of Canon 915 of the Code of Canon Law:

Those ....  obstinately persevering in manifest grave sin, are not to be admitted to Holy Communion. 

The key words here are, firstly, persevering. For the person who is re-married, their situation is one that persists and is not going to foreseeably end, it is one in which they perservere, continue. Secondly, the word manifest. Their situation is a visible situation, one that can be seen in the public record and practice of life, that can give scandal in the technical sense of the word.  And the third word is grave (sin). Divorce and re-marriage constitutes what is termed "grave matter", and this not only because of the injustice it represents to the nature of marriage seen in its human dimension but also because it denies its irrevocable representation of the union of Christ and his Church, that is, the love of God towards mankind.

In other words, it is the "objective situation of sin", to use the words of n.305, that is the ground for not admitting the divorced and remarried to Holy Communion, and this remains in place independently of any judgement or discernment that might be made with regard to mitigating factors.

The teaching of the Catechism of the Catholic Church on mortal sin is found in n.1855 ff (my italics added):

Mortal sin destroys charity in the heart of man by a grave violation of God's law; it turns man away from God, who is his ultimate end and his beatitude, by preferring an inferior good to him. .... Mortal sin, by attacking the vital principle within us - that is, charity - necessitates a new initiative of God's mercy and a conversion of heart which is normally accomplished within the setting of the sacrament of reconciliation.... For a sin to be mortal, three conditions must together be met: "Mortal sin is sin whose object is grave matter and which is also committed with full knowledge and deliberate consent.".... Mortal sin requires full knowledge and complete consent. It presupposes knowledge of the sinful character of the act, of its opposition to God's law. It also implies a consent sufficiently deliberate to be a personal choice.

It is clear from this that a person who is divorced and remarried might be in an objective situation of sin - and therefore be unable to receive Holy Communion - but not in a situation of mortal sin. Equally, if the conditions of knowledge and consent are met, they might be both in a situation of objective sin and unable to receive Holy Communion for that reason, and in a situation of mortal sin and unable to receive Holy Communion for that reason too (cf Canon 916), though it is unlikely to be publicly visible.

When Amoris Laetitia n.305 refers to the possibility that someone living in an objective situation of sin can nevertheless be "living in God's grace" it is referring to the first of these two possibilities. Charity might be wounded in that case, but it is not destroyed as would be the case in the second of the two possibilities.

This then makes complete sense of Amoris Laetitia n.306:

In every situation, when dealing with those who have difficulties in living God's law to the full, the invitation to pursue the via caritatis must be clearly heard. Fraternal charity is the first law of Christians ....

Where "the vital principle within us - that is, charity -" has not been broken, then the pastoral discernment of the particular situation should arrive at that style of charitable engagement that is appropriate to the individual situation and which will lead to a growth in grace. It is this growth in charity that is the first object of pastoral discernment and accompaniment.

It also makes sense of the reference in the preceding n.303 to the role of conscience, suggesting that it will firstly recognise the objective wrong of the situation and then prompt towards a progress along the way of charity:

...conscience can do more than recognise that a given situation does not correspond objectively to the overall demands of the Gospel. It can also recognise with sincerity and honesty what for now is the most generous response which can be given to God, and come to see with a certain moral security that is what God himself is asking ...

Much of the comment that is critical of Amoris Laetitia, and of these paragraphs in particular, lazily conflates the "objective situation of sin" to being one of "mortal sin", and with similar laziness identifies the mortal sin as the canonical ground for the Church's discipline of refusing the divorced and remarried Holy Communion. Having done these two, it is then natural to accuse these paragraphs of Amoris Laetitia and their urging of pastoral discernment - utterly  inaccurately - of having overturned the Church's discipline on Holy Communion for the divorced and remarried.