One game I played in my childhood may be called “throw mud balls” in English.

Your original was not wrong but adding called makes it more clear that you are doing a direct translation.

The game was played on slightly rainy days.

Lightly is used correctly here but it sounds more natural to say slightly.

The best branches were about 50 to 60 centimeters long and the diameter of the middle of them was around 5 to 6 millimeters, which was considered the best branch to play the game.

By putting "which was considered the best branch to play the game." at the end it feels redundant.

WhenAfter players finish preparing their branches were ready, players, they would grabbed a handful of mud and shaped it into many small balls of suitableabout golf ball size.

How were the branches made ready? Do they need to be stripped or prepared in some way? While this is not required for your story, starting a sentence with the phrase "When the branches were ready" makes it feel like there was a previous step that we missed. "suitable size" feels redundant unless there is a specific size you can reference.

At this stage, the mud should have moderatstill be a little moisture; otherwise, it couldn't be made into small balls.

"moderate moisture" sounds like you are doing a weather report or a more formal account. If that is your intent then it is good.

Attach a smallmud ball to the thin end of the branch tightsecurely, hold the thick end of the branch, and then try hard to throw the ball as far as you can.

"tightly" is usually used in a different context like making a knot. Try using a different word there. "try hard" is a bit clunky here. You are already throwing as far as you can so it can be assumed that you are trying hard.

One game I played in my childhood may bewas called “throw mud balls” in English.

The game was played on lightly rainy days.

The branches were about 50 to 60 centimeters long, and the diameter of the middle of thembranch was around 5 to 6 millimeters, which was considered the best branch tofor playing the game.

When the branches were ready, players grabbed a handful of mud and shaped it into many small balls of suitable size.

At this stage, the mud should have moderate moisture; otherwise, it couldn't be made into small balls.

When the branches and small mud balls were all ready, the gamwe could start the game.

Attach a small ball to the thin end of the branch tightly, hold the thick end of the branch, and then try hard to throw the ball as far as you canpossible.

Whoever throws the ball farthest is the winner.

That is all for this journal.

Throwing Mud Balls

One game I played in my childhood may be called “throwing mud balls” in English.

The game was played on days with lightly rainy days.

It is okay before, but this sounds better.

The branches were about 50 to 60 centimeters long, and thea diameter ofin the middle of them was around 5 to 6 millimeters, which was considered the best branch to playthickness for the game.

AYou would attach a small ball tightly to the thin end of the branch tightly, hold the thick end of the branch, and then try hard to throw the ball as far as you can.

"hard" is not necessary because you already say "as far as you can".