The best description of the effect of the expression "Woe's me" is that It heightens the sense of loss in the poem.
Explanation:
The expression Woe's me is used to feel sorry for one's self in the line "Woe's me, I shall be lonely / When I can feel no longer /The impatience of their wings! this word emphasizes the feeling of loneliness and loss of the person, as it gives a darker and painful mood to the message, this is sometimes considered too much since it is often used when the conflict is not that big.