I forgot about some training I had to do that day.
I had some training I had to do that day that completely slipped my mind.
「その日研修が入っていたことを忘れていた」は次のような言い方ができます。
ーI forgot about some training I had to do that day.
to forget about ... I had to do that day を使った言い方。
ーI had some training I had to do that day that completely slipped my mind.
to have ... I had to do that day that completely slipped one's mind を使った言い方。
例：
Sorry I had to cancel our last lesson. I forgot about some training I had to do that day.
「前回のレッスンをキャンセルしてしまいすみません。あの日研修が入っていたのを忘れていました」
ご参考まで！
カナダ 
