A tanárnő tollat ad nekem, de én inkább ceruzával írok.

Yes.

• tanár = teacher
• tanárnő = female teacher

So A tanárnő means the female teacher. In natural English, you will often just translate it as the teacher, unless the fact that she is a woman matters.

In Hungarian, tanárnő is very common in school contexts when referring to a woman teacher.

A is the Hungarian definite article, like English the.

Hungarian has two forms:

• a before a consonant
• az before a vowel

Since tanárnő starts with t, you use a:

• a tanárnő = the teacher

Because tollat is the accusative form, and the accusative marks the direct object.

• toll = pen
• tollat = pen as the thing being given

In this sentence, the teacher is giving a pen, so pen is the direct object:

• A tanárnő tollat ad nekem.

The ending -t is the usual accusative marker in Hungarian.

Nekem means to me and is the dative form of én.

Here it marks the recipient of the action:

• the teacher gives a pen
• to me

So:

• nekem = to me / for me

This is different from engem, which means me as a direct object.

• nekem = to me
• engem = me

In this sentence, nekem is correct because the teacher is not giving me; she is giving a pen to me.

The ending -val/-vel means with.

So:

• ceruza = pencil
• ceruzával = with a pencil

In the sentence, the pencil is the instrument used for writing:

• ceruzával írok = I write with a pencil

This is not a direct object, so you do not use the accusative -t here.

Also notice this spelling change:

• ceruza
  • -val → ceruzával

The final a becomes á before the suffix.

This is because Hungarian has indefinite and definite verb conjugation.

Here the sentence uses the indefinite conjugation:

• ad = he/she gives
• írok = I write

Why indefinite?

• tollat is an indefinite object: a pen, not the pen
• in ceruzával írok, there is no definite direct object at all

If the object were definite, you would usually use the definite conjugation:

• a tollat adja = he/she gives the pen
• a levelet írom = I am writing the letter

So the forms in your sentence are exactly what you would expect.

That is true: Hungarian usually leaves out subject pronouns because the verb ending already shows the person.

But én is often included for contrast or emphasis.

Here you have:

• de én inkább... = but I would rather...

So én is there to highlight the contrast between:

• what the teacher gives
• what I prefer to use

Without én, the sentence would still be grammatical, but the contrast would be weaker.

Inkább means rather, preferably, or instead.

In this sentence it shows preference:

• én inkább ceruzával írok = I would rather write with a pencil

It is a very common word when you want to contrast two options:

• kávét kérek inkább = I’d rather have coffee
• inkább otthon maradok = I’d rather stay home

Because Hungarian often leaves indefinite singular nouns without an article when English would use a/an.

So all of these are natural:

• tollat ad = gives a pen
• ceruzával írok = I write with a pencil

You can add egy if you want to make one more explicit or slightly more concrete:

• egy tollat ad nekem = she gives me a pen / one pen

But in many everyday sentences, Hungarian simply does not need egy where English needs a/an.

It can mean either, depending on context.

Hungarian present tense often covers both:

• I write
• I am writing

In this sentence, the idea is more like a general preference:

• I write / prefer to write with a pencil

If you want to make it clearly habitual, Hungarian might use something like szoktam. If you want to make it clearly happening right now, context or a time word like most often does the job.

Hungarian word order is more flexible than English word order. It is strongly influenced by topic and focus, not just by subject-verb-object order.

In your sentence:

• A tanárnő sets up the topic
• tollat ad nekem tells what she does
• de én inkább ceruzával írok gives a contrasting statement

A very important idea in Hungarian is that the part right before the verb often gets strong emphasis or focus. In the second clause, ceruzával is in a prominent position before írok, which fits the idea that the speaker prefers that instrument.

So the sentence is not random at all; its word order helps express the contrast and preference.

They can often be moved around, but the emphasis usually changes.

Hungarian is flexible, but different orders do not always feel equally neutral.

For example:

• A tanárnő tollat ad nekem = natural, neutral
• Nekem ad tollat a tanárnő = stronger emphasis on to me
• Én inkább ceruzával írok = natural contrast
• Ceruzával írok inkább = possible in some contexts, but different rhythm/emphasis

So yes, movement is possible, but it is better to learn the original version first and treat word order changes as changes in emphasis, not just style.