We couldn't bear to hear the pitiful cries of the sick child.

Literal

Sick [possessive-の] child [possessive-の] pitiful crying [object-を] hearing [topic-のは] us [for-には] was-unbearable.

痛ましい泣き声 means 'pitiful/heartrending cries.' 聞くのは...耐えられなかった nominalizes 'hearing' as the topic and pairs it with 耐えられない ('unbearable'). The potential negative 耐えられなかった expresses inability to endure.