My boss gave me a candy called “Gumitzel”.”

The period should be inside the quotation marks.

It was shaped like a traditional small pretzel 🥨 but made of jelly coated with a hard and, thin sugar layelayer of sugar.

Add the article "a" before "hard."
Add a comma between "hard" and "thin" because they are coordinate adjectives.
"Layer of sugar" is clearer and more natural than "sugar layer."

We call jelly “gumi” or “gumi candy” in my country,; that is where itsthe name (“Gumitzel)” comes from.

Use a semicolon instead of a comma to separate two independent clauses.
“The name” sounds more natural than “its name” in this context.

Actually, I don’t like jelliesy very much, but they areit is very popular, and most convenience stores always have at least 10 kinds of “gumi”s on their shelves.

Add a comma after “Actually.”
“Jelly” is usually uncountable, so “jellies” → “jelly.”
“It is very popular” fits better here.
“Kinds of gumi” — no plural “s” after “gumi” since it’s treated like a mass noun here

So I didn’t eat it myself and; instead, I took it home as a souvenir for my son.

The original sentence is grammatically correct, but by adding words or changing the structure slightly, it sounds more natural and flows better when spoken or read.
Adding “instead” clarifies the reason and improves flow.

He loved it and ate ithem slowly, by breaking ithem into small pieces and, picking them one by one, then cautiously put itting them into his mouth.

Keep pronouns consistent: use “it” for singular candy (Gumitzel).
Use parallel gerunds “breaking,” “picking,” “putting” for smoother flow.
Use “them” for the small pieces, since the candy is broken into multiple pieces.

My boss gave me asome candy called “Gumitzel”.

From the last sentence it seems you got more than one, so "some" is better than "a" here, as "a" would imply he give you a single piece of candy.

It was shaped like a traditional small pretzel 🥨 but made of jelly coated with a hard and thin sugar layer.

We call jelly “gumi” or “gumi candy” in my country,; that is where itsthe name (of Gumitzel) comes from.

The jelly inside wais extremely soft and melty, which contrasteds nicely with the texture of the crispy surface.

To say "the jelly was" would imply you ate the jelly you were given and are talking about that specific jelly. But later on you say you didn't. If you're just describing the characteristics of that kind of jelly in general, it's better to stick to the present tense, as that's still true about the variety of jelly regardless of what happened to the specific one you were given.

So I didn’t eat ithem myself and instead took ithem home as a souvenir for my son.

"instead" helps add to the contrast between the first clause and last one. It's correct without it also, but it makes it sound more natural to include it.

He loved ithem and ate them slowly, by breaking them into small pieces and picking them one by one, then cautiously putting it into his mouth.

It was shaped like a traditional small pretzel 🥨 but made of jelly coated with a hard and thin sugar layer.

We call jelly “gumi” or “gumi candy” in my country,; that is where its name (Gumitzel) comes from.

Actually, I don’t like jellies very much, but they are very popular and most convenience stores always have at least 10 kinds of “gumi”s on their shelves.

He loved it and ate ithem slowly, by breaking ithem into small pieces and picking them one by one, then cautiously put itting them into his mouth.

Mind the grammatical number. Initially, you should use "it" since there is only one piece of candy. Afterwards, you should use "them" because you're referring to the multiple smaller pieces of candy.

It was shaped like a traditional small pretzel 🥨 but was made of jelly coated with a hard and thin sugar layer.

We call jelly “gumi” or “gumi candy” in my country, t. That is where its name (Gumitzel) comes from.

Actually, I don’t like jellies very much, but they are very popular and most convenience stores always have at least 10 kinds of “gumi”s on their shelves.

So, I didn’t eat it myself and took it home as a souvenir for my son.

He loved it and ate them slowly, by breaking them into small pieces and picking them one by one, then cautiously put itting them into his mouth.