My Child's Education of Child

This sounds more natural in the US.

My daughter entered elementary school at this (past) April.

Now she is in thehas a long summer vacation.

Your sentence is technically correct but this is more how'd we say it in the US.

However, oOn weekdays, both I and my wifehowever, my wife and I both work, so we can't take care of her.

or, even better:

On weekdays, however, my wife and I both work, so we can't look after her.

We leave her in the care of a "gakudo"."

"Gakudo" is like aschool but it's less formal school.

Many children in the samesimilar situations study and play togathere togethere.

I don't know much about "Gakudo" well," but it's sounds fun.

I'm not sure whether to capitalize, "Gakudo/gakudo," or not.

MThis April, my daughter entered elementary school at this April.

There's nothing wrong with putting "this April" at the end, but I think it sounds better if it starts the sentence.

Now she is ion ther long summer vacation.

However, on weekdays, both I and my wife and I work, so we can't take care of her.

In English, when we are listing people, it's usually seen as polite (or proper) to put "I" or "me" as the last thing in the list. But your sentence is fine on its own!

We leave her in the care of a "gakudo".

'of "gakudo"' would imply that gakudo is a person.

"Gakudo" is like a less formal form of school.

Many children in the same situation study and play togaether there.

I don't know much about "Gakudos" well, but it'sthey sounds fun.