Harvest Manying Sweet Potatoes

This activity was underwent by theShe did this at kindergardten, and it is thea popular event for kids in Japan.

The first half of this sentence sounds quite strange to me. Rephrasing it like this feels most natural to me.
Since there are, most likely, other events which are popular, it is most appropriate to use "a" instead of "the" here.

She told me that she enjoyed it very wellmuch.

Rephrasing to be more natural.

Harvested potatoes were cooked My wife baked the potatoes into sweet potato cakes by my wife.

Generally it is best to avoid using the passive voice like this (sentences with a structure like: "[object] was [verb]ed by [subject]"). It is best to rephrase in the active voice (sentences with a structure like: "[subject] [verb]ed [object]").
Here since we already know that the sweet potatoes were harvested it isn't necessary to repeat the information, so we can just call them "the potatoes".

But some potatoes have still remained yetthere are still some potatoes left over.

Rephrasing to be more natural.

WeThey should use theselast for a few days.

Guessing at your meaning here - I guessed that you meant that they will probably stay good for a few days, over which you will probably eat them. It could also be "we should use these up over the next few days" if you mean that you want to make a conscious effort to use them up.

Harvesting Many Sweet Potatoes

"Many" sounds a bit weird here. I would say "Harvesting Sweet Potatoes" sounds more natural of a title to me. Then you can later elaborate that it was "many sweet potatoes" that were harvested.

This activity was underwentdone by the kindergardteners, and it is thea popular event for kids in Japan.

Despite "kindergarten" being pronounced as "kinder garden" it's not spelt that way at all. Isn't spelling in English fun!?!

An alternate suggestion:
"This activity was preformed by the kindergarteners, and it is a popular event for kids in Japan."

"was underwent" doesn't sound correct, if you want to use underwent, you could use it like the following:
"The sweet potatoes underwent harvesting by the kindergarteners." (Though this sounds overly formal, like it's being presented by a news report)
"The roads underwent construction."
"The boy underwent a change."

The key is that a verb has to follow "underwent" for it to sound correct. Also, it sound like a really formal word that's used more for stating things in a factual manner.

She told me that she enjoyed it very wellmuch.

Alternate suggestion:
"enjoyed it a lot."

HThe harvested potatoes were cooked into sweet potato cakes by my wife.

"cooked into" sounds a little unnatural, I usually hear/say the phrase "made into".

But some potatoes have still remained yetthere were still some potatoes remaining.

This sentence structure is hard. I would say a good take away from this correction is that,

"still" and "yet" are synonyms (but have different word order), and generally shouldn't be used in the same sentence.

WThese should use theselast for a few day.

This correction is a very common phrase in English (but maybe not what you were trying to say). Some other suggestions may be:

"We should use these over the following days."
"We need to use these over the next few days (or else they'll go bad)."
"We will use them over the next few days."

My daughter told me that she harvested manya lot of sweet potatoes today.

In this context, "a lot of" is more appropriate than "many".

This activity was underwentdone by ther kindergardten class, and it is thea popular event for kids in Japan.

She told me that she enjoyed it very wella lot.

HarvestedThe sweet potatoes she harvested were cooked and made into sweet potato cakes by my wife.

As I recall, the cooking and cake-making are separate steps, so it's better to be explicit.

But some potatoes have still remained ye, we still have some sweet potatoes left.

"have XXX left" is the way to say that you were not able to use up your supply of XXX.

We should usebe using these forup in a few days.

Corresponding to "have XXX left", "use XXX up" is how you describe what you'll be doing with the rest of the XXX.