Today’s lecture was about Ayu fish.

Ayu is the most popular fish of freshwater fish in Japan.

(noun of noun) and (adjective noun) both translate to の in Japanese, but in English they're not interchangeable

But othersome people cannot eat this fish.

BThat's because, Ayu has a distinct smell like cucumbers or watermelon.

In this time, we can see thatFor now, we can find the fish in the supermarket tountil the end of summer.

Had to take a guess as to the intended meaning, the original wasn't clear.

Ayu is traditional ofcuisine in Japan, Thailand and China.

In Japan more productionalyit is generally (fished / farmed) in Gifu prefecture by inland aquaculture.

The difference between fishing and fish farming is that you'd say "fishing" if the fish was going to be there anyway, regardless of what the fisher does, while "fish farming" implies that the people doing it are also involved in breeding or introducing the fish to the environment.

Since you mention aquaculture, I assume it's specifically a fish farming setup.

In this prefecture hathere is the Research Institute for Fisheries and Aquatic Environments and the Hatchery of Ayu spises (?).

Not sure what Ayu spises is meant to be, is this a specific variety of the Ayu fish?

In this area has athere are two rivers, thoseeir names are "Nagara Gawa" and "Kiso Gawa" ¶
. Those river are more important for Ayu production.

"A" is used for singular things. "Two rivers" is a plural phrase, so there is no "a" used.

You can use either of:

"In this <X> there are"
"This X has"

But you can't combine them into "In this X has".

"more" needs something to compare to (usually with "than"). You can say "this river is more important for Ayu production than other rivers", but you can't say "this river is more important for Ayu production" on its own.

The nNext week, I will go to to this prefecture for eat Ayu.

Today’s lecture was about Aayu fish.

Ayu is the most popular fish of freshwater fish in Japan.

But many other people cannot eat this fish.

BThis is because, A ayu has a distinct smell like cucumbers or watermelon.

In this time,Now we can see thait in the supermarket through to summer.

Ayu is traditional ofin Japan, Thailand and China.

In Japan more are productionalyed in Gifu prefecture by inland aquaculture.

In this prefecture hais the Research Institute for Fisheries and Aquatic Environments, and Hhatcheryies of Aayu specises.

In this area has aare two river, those names ares, named "Nagara Gawa" and "Kiso Gawa." ¶
Those rivers are more important for Aayu production.

The nNext week, I will go to this prefecture forto eat Aayu.

Today’s lecture in the university

Depending on where in the world a person lives, we might say college instead of university. For example in the US we might say "Today's college lecture"

Today’s lecture is about the Ayu fish.

Can also say "Today's lecture will be about the Aya fish."

Ayu is the most popular fish of freshwater fish in Japan.

There are a few ways to say this: "Ayu is the most popular of the Japanese freshwater fish." or "Ayu is the most popular fish of the freshwater fish species in Japan."

But otherSome people cannot eat this fish.

Cannot eat is different from don't like to eat. Alternately, can say "Some people do not like to eat this fish."

BThis is because, Ayu has a distinct smell likesimilar to cucumbers or watermelon.

Cucumbers and watermelon don't really smell the same (to me). Should the "or" be an "and"?
This sentence could also come before, "Some people cannot eat this fish" for example. "Ayu has a distinct smell similar to cucumbers or watermelon. Because of this fact, some people don't like to eat this fish."

In this time, we can see thate summer season, we can find this fish in the supermarket to summer..

Or..... "In the summer season, we can buy this fish in the supermarket."

Ayu is traditional of Japan, Thailand, and China.

Oxford comma should be used.

In Japan morese productionaly in Gifu prefecture by of the fish is primarily done using inland aquaculture. in Gifu.

It's not completely clear what is more important in this sentence, the production or the inland aquaculture. Hopefully this correction helps.

In this prefecture has tThe Research Institute for Fisheries and Aquatic Environments and Hatchery of Ayu spises.ecies is located in the Gifu prefecture.

In this area has a two river, those names areThere are two rivers in this area, they are called "Nagara Gawa" and "Kiso Gawa" ¶
Thoese rivers are more important for Ayu production.

The nNext week, I will go to this prefecture foram going to Gifu to eat Ayu.