I fotgot to take the key went away the home in the morning

是有问题
应改作 I forgot to take the key when I left home in the morning.
其中when I left还可改作when leaving

I forgot taking the key when i left home this morning.

应该这样说

有 在went away the home 那
如果你想表达的是早上出门忘带钥匙了的话
I forgot to take the key as I went out home this morning就可以

I forgot MY keys this morning when I left home.如果你想表达家门钥匙的话,用my,不要用the!

I forgot taking the key when i left home this morning.