How to write it？
ハンカチ
ハンケチ
ハンカチーフ
手巾 、手帛、手帕、手布
半巾 、半帛
When writing handkerchief in Japanese, You can use Katakana ハンカチ.
ハンカチーフ is the Japanese like pronunciation of ハンカチーフ.
ハンカチ and ハンケチ are a short version of ハンカチーフ.
Nowadays, we use ハンカチ in daily conversations. You hardly see ハンケチ, ハンカチーフ and kanji versions in everyday life. You can see them in novels, especially which written in before the Taisho era.
Advanced Tips
手巾 , 手帛, 手帕, 手布 are made up of a kanji meaning hand and another kanji meaning cloth. I suppose they include a kanji 手=hand because handkerchiefs are associated with 手ぬぐい which is a Japanese traditional towel used to wipe hands or sweat.
半巾 , 半帛 are made up of a kanji meaning half and another kanji meaning cloth. I suppose they include a kanji 半=half because 半 and ハン of ハンカチ are the same sounds “han”. Or possibly, they could come from the half in 手ぬぐい.
