Describes eyes that are not fully open, often due to drowsiness, relaxation, or suspicion. Similar to 薄目 (うすめ) but 半目 is more about the state of being half-open rather than squinting.
He was watching TV with half-open eyes, looking sleepy.
The cat has its eyes half-closed, looking comfortable.
Compound of 半 (half) and 目 (eye). The reading はんめ follows standard on'yomi.