「居残り」の英訳は色々あります。
文脈が分かりませんのでどれが適切なのか分かりませんが、学校で悪い事をして、そして先生に放課後居残りを命じられた場所は Detention と言います。Kept behind と言い方もあります。
例えば
-I got detention because I got into a fight.
-I was kept behind after school because I got into a fight.
-I was told to stay behind after school because I got into a fight.
仕事の場合は居残りを命じられたら Overtime と言います。残業と同じです。
例えば
-I had to do/work overtime today.
手伝ってと頼まれた場合は学校でも仕事でも Stay behind (and help)もしくは help だけになると思います。
例えば
-My teacher asked me to help her after school.
-My teacher asked me if I could stay behind and help her with some things after school.
-My boss asked me to help him with a couple of things after work.
-My boss asked me if I could stay behind and help him with a couple of things after work.
彼は毎週、居残りをしている- He stays behind every week (直訳)
大変申し訳ございませんが、文脈が分からないと適切な翻訳出来ません。
例文
He always stays behind after work.
He always stays behind after school to study.
この場合は、命令や頼みはなく、自分自身で居残りをすることを決めました。
イギリス 
