Barnet kan lege frit i haven.

In Danish, the definite article is usually added to the end of the noun instead of being a separate word in front.

So:

• et barn = a child
• barnet = the child

The ending -et is the definite ending used with many et nouns, also called neuter nouns.

Because have is an en noun, not an et noun.

So:

• en have = a garden
• haven = the garden

Danish has two grammatical genders for nouns:

• en nouns
• et nouns

The definite form is usually:

• -en for en nouns
• -et for et nouns

That is why you get barnet but haven.

Unfortunately, you usually have to learn the gender together with the noun.

A good habit is to memorize nouns as whole units:

• et barn
• en have

There are some patterns, but they are not reliable enough to guess every time. Most Danish nouns are en nouns, but many very common words are et nouns, so it is best to learn them individually.

After a modal verb like kan, the next verb stays in the infinitive form.

So:

• kan lege = can play
• not kan leger

This is similar to English:

• can play
• not can plays

Common Danish modal verbs include:

• kan = can
• vil = will / want to
• skal = shall / must / have to
• må = may / be allowed to
• bør = ought to

After these, the main verb does not take -r.

Because after a modal verb, Danish normally does not use at.

So you say:

• Barnet kan lege
• not Barnet kan at lege

This is much like English:

• The child can play
• not The child can to play

It can suggest more than one of those, depending on context.

In Barnet kan lege frit i haven, kan could mean:

• the child is able to play freely in the garden
• the child is allowed to play freely in the garden
• it is possible for the child to play freely in the garden

Context tells you which reading is strongest.

If you want to make permission clearer, Danish often uses må:

• Barnet må lege i haven = the child may / is allowed to play in the garden

Here, frit is being used adverbially, meaning freely.

It comes from the adjective fri = free. Its neuter form is frit, and that same form is often used as an adverb.

So:

• fri = free
• frit = freely / free in neuter form

Compare:

• Barnet er frit = the child is free
• Barnet kan lege frit = the child can play freely

So in this sentence, frit describes how the child can play.

Because i is the normal preposition for being in an enclosed or defined space/area, and a garden is usually treated that way in Danish.

So:

• i haven = in the garden

Using på here would normally sound wrong. Danish and English prepositions do not always match perfectly, so it is best to learn common combinations like:

• i haven
• i huset = in the house
• på bordet = on the table

Yes, but the meaning or emphasis may change a little.

The sentence Barnet kan lege frit i haven is a very neutral word order.

You may also hear:

• Barnet kan frit lege i haven
• I haven kan barnet lege frit

A few important points:

• Danish is a V2 language, which means the finite verb usually comes in the second position in main clauses.
• If you move something to the front, the verb still has to stay second.

So:

• Barnet kan lege frit i haven
• I haven kan barnet lege frit

Both are correct, but the second one puts extra focus on in the garden.

It normally goes after the finite verb.

So:

• Barnet kan ikke lege frit i haven = the child cannot play freely in the garden

That placement is very common in Danish main clauses:

• subject + finite verb + ikke
  • rest

So here:

• Barnet = subject
• kan = finite verb
• ikke = not
• lege frit i haven = rest of the clause

Not in all situations.

Lege is the usual word for:

• children playing
• playing in a general, playful sense
• pretend play

But Danish often uses spille for some other kinds of play, especially:

• playing a game
• playing a sport
• playing a musical instrument

For example:

• Barnet leger i haven = the child is playing in the garden
• Barnet spiller fodbold = the child is playing football
• Hun spiller klaver = she plays the piano

So lege is exactly the right choice in this sentence.