Hän ei itke pitkään, koska äiti tuo uuden jäätelön.

Hän can mean either he or she. Standard Finnish does not normally mark gender in the third-person singular pronoun.

So:

• hän = he/she
• context tells you which one is meant

In everyday spoken Finnish, people often use se for humans too, but in standard written Finnish, hän is the normal choice.

Finnish makes negation with a separate negative verb plus the main verb.

Here:

• ei = the negative verb for he/she
• itke = the main verb itkeä in its special negative form

So hän ei itke literally works like he/she not cry.

A very important point: after ei, the main verb does not take the normal personal ending.

Compare:

• hän itkee = he/she cries
• hän ei itke = he/she does not cry

Not ei itkee.

Pitkään means for a long time.

The basic adjective is:

• pitkä = long

But here Finnish needs an adverbial time expression, not just the adjective. So:

• itkeä pitkään = to cry for a long time

It is best to learn pitkään here as a set expression meaning for long / for a long time.

A very common alternative is kauan:

• Hän ei itke kauan.
• Hän ei itke pitkään.

Both can mean roughly He/She won’t cry for long.

Finnish does not have articles like English a/an and the.

So a noun can be understood as definite or indefinite from context.

For example:

• äiti can mean mother, the mother, or mom
• jäätelön can mean an ice cream or the ice cream, depending on the situation

This is normal in Finnish. You do not need to add a separate word for a or the.

Because Finnish often leaves that kind of relationship clear from context.

So äiti can naturally mean:

• mom
• the mother
• his/her mother

If you really wanted to spell out his/her mother, you could say:

• hänen äitinsä

But that is often unnecessary if the context already makes it obvious.

Yes, tuo can also be a demonstrative meaning that, but here it is a verb.

In this sentence:

• tuo = brings
• base form: tuoda = to bring

So:

• äiti tuo = mother brings / mom brings

Finnish present tense can also refer to the near future, depending on context. So äiti tuo uuden jäätelön can mean:

• mother brings a new ice cream
• mother is bringing a new ice cream
• mother will bring a new ice cream

Because this noun phrase is the object of the verb tuo.

In an affirmative clause, a singular total object often appears in a form that looks like the genitive:

• uusi jäätelö = nominative, dictionary-style form
• uuden jäätelön = object form here

Both words change because the adjective agrees with the noun:

• uusi → uuden
• jäätelö → jäätelön

So äiti tuo uuden jäätelön means that the mother brings one whole new ice cream.

Because the negation belongs to the main clause, not to the because-clause where the object appears.

Structure:

• Hän ei itke pitkään = main clause, negative
• koska äiti tuo uuden jäätelön = subordinate clause, affirmative

The object uuden jäätelön belongs to äiti tuo..., and that clause is not negative.

This is why uuden jäätelön is correct here.

For comparison:

• Äiti tuo uuden jäätelön. = Mother brings a new ice cream.
• Äiti ei tuo uutta jäätelöä. = Mother does not bring new ice cream / a new ice cream.

So the partitive uutta jäätelöä would appear if that clause itself were negative.

Because koska introduces a subordinate clause, and Finnish normally separates subordinate clauses with a comma.

So this punctuation is standard:

• Hän ei itke pitkään, koska äiti tuo uuden jäätelön.

This is one difference from English punctuation, where comma use before because can be more limited.

Finnish word order is fairly flexible, but the given version is the most neutral and straightforward.

Current order:

• Hän ei itke pitkään, koska äiti tuo uuden jäätelön.

A different order is also possible, for example:

• Koska äiti tuo uuden jäätelön, hän ei itke pitkään.

That still means the same basic thing, but the emphasis changes slightly. The original sentence sounds very natural as a plain statement.