My feeling is that reincarnation is not a part of Plan A for Christians - it is not normal. But neither is it excluded nor impossible - it sometimes happens.
But I suspect we are not supposed to focus or dwell upon it, because to do so would be to the detriment of Plan A.
So, just in brief, when might reincarnation be a possibility?I can think of two, almost opposite, situations.
1. When an exceptionally Good person returns to earth as a teacher or prophet. I suspect that one reason that reincarnation has been, mistakenly, taught as a standard necessity is that some of the greatest religious teachers and prophets were themselves reincarnated.
For these teachers and prophets, I presume that mystical introspection told them that they personally had been reincarnated - which was true - but they mistakenly generalized this to apply other people: they made it a principle - which was an error.
2. When a soul comes to earth not only to experience incarnation and death - which is the universal necessity for spiritual progression - but also to have some particular kind of experience and this experience did not happen due to some unforeseen and accidental exercise of agency on the part of others: perhaps (sometimes) if a foetus is aborted, or someone (perhaps a child) is murdered before they have been able to have the experience which was an important reason for their incarnation,
(These are just suggestions; and we, in mortal life, would not necessarily know who were these people liable to reincarnation. But this may be a clue as to why most murders are regarded as a particularly bad sin)
In such circumstances, I would imagine that God might allow a soul to be reincarnated, and have another try - a second, or more than second, chance - if that was what the person wanted.
Because reincarnation (assuming it does happen), like incarnation in general, is and must (surely?) be the act of a volunteer: a choice.
The idea of a God who uses cycles of reincarnation either to torment or to punish souls is (surely?) incompatible with the loving Father God of Christianity; and Mormonism in particular is clear that incarnation is not compelled but is an act of free agency.
In sum, there may be, I think have been, exceptional situations in which reincarnation - another loop through mortal experience, or more than one such loop is permitted to volunteers; either for the good of that soul or for the good of other souls.
But reincarnation is Plan B (or perhaps Plan C, D or E...) - reincarnation is not the Plan of Salvation and Spiritual Progress as it applies to most people, most of the time.