Shouldn't there be a second הן in this sentence?
הן התאוששו אבל הן עדיין עייפים?
Why is this wrong: (mine had the female version too.) The entire exercise including the tips says that in the past you don't need pronouns because the past tense is specific.
התאוששו אבל הם עדיין עייפים
As I wrote above - third person pronouns (he, she, they) are required.
Please check the exercise; the word bank I was given had only feminine plural.
hem heetosheshú aval adáin ayefim
feminine with audio: https://forum.duolingo.com/comment/28534830
Is the pronoun "הם" really mandatory in this sentence?
It is; generally speaking third person pronouns are required in Hebrew.