To cut one's hand (literally), i.e., to injure the hand by cutting the skin.
包丁で手を切ってしまい、病院で縫合した。
I cut my hand with a kitchen knife and had it stitched at the hospital.
工場でガラスの破片に触れて手を切った。
At the factory I touched a shard of glass and cut my hand.
紙で指先を少し切ったが、すぐに治った。
I slightly cut my fingertip with paper, but it healed quickly.
To cut one's connections with someone; to break off relations or cease dealings with someone or something (idiomatic).
彼はトラブルを起こしたグループと手を切ることにした。
He decided to cut ties with the group that caused trouble.
会社の不正に関わった人とは早く手を切ったほうがいい。
It's better to sever ties early with people involved in the company's wrongdoing.
危険な交友関係とは早めに手を切るべきだ。
You should break off dangerous friendships early.