A somewhat dated or child-oriented word for a handkerchief; modern Japanese more commonly uses ハンカチ.
祖母はいつも鼻拭きを持ち歩いていた。
My grandmother always carried a handkerchief with her.
Compound of 鼻 (hana, 'nose') and 拭き (fuki, 'wiping'), literally 'nose-wiper'.