The speaker freely acknowledges that the death of a child by fire in London is a tragic event. He desperately wants to mourn for her but refuses to do so, not out of callousness or indifference, but because his focus is elsewhere.
In this poem, the speaker is concerned with death in general rather than the death of a specific individual. His focus is firmly upon mankind, of which he, the child, and everyone else is a part.
In the third stanza, the speaker steadfastly refuses to "murder" the child's humanity, her membership of mankind, by elegizing her innocence and youth. It is as a member of mankind that the speaker sees the child, not as an individual.
Indeed, the word mankind and what it represents is so important to the speaker that he repeats it. In the first stanza, we are told that mankind is made out of darkness—the darkness from which we all come and to which we will all return when we die.
That is the bigger picture here. And the speaker feels that by mourning the death of the child, he will be losing sight of the meaningfulness of death for mankind in general and that, furthermore, by sentimentalizing death, he will be sinning against the truth as he understands it.