It needs to be a possessive (in the first part of the sentence it is 'the child's' which indicates possession). In the second part, 'the child's' needs to be replaced by a possessive pronoun. Which means that 'their' is the correct choice.
Unfortunately, although 'their' is commonly used these days in this circumstance (to avoid specifying a sex), it sounds a bit strange since 'child' is singular and, to be strictly grammatical, the pronoun should be his/her or (possibly) its.
However, as I say 'their' is often considered acceptable nowadays.
They can, Karen, but you have to decide what you want to say.
Although I don't like "their" after "a child", I hear this use of they/their frequently. People who use it think that the old singular pronouns like "his" or "her" are not fair. After all, children have two genders.
"The" is certainly possible, but it sounds somewhat detached and scientific to me.
If I needed to write this sentence, I'd probably choose the clumsy "that child" because I really dislike "their" after a singular noun: ...but also that child's psychological and social well-being.