A punção com a agulha deixou um hematoma.

Not exactly.

Punção is a more technical/medical word meaning a puncture, needle insertion, or needle stick. It focuses on the act of piercing with a needle.

Injeção means an injection specifically—the act of injecting a substance into the body.

So:

• punção = puncture / needle insertion
• injeção = injection

In some real-life situations, the same event may involve both, but the words are not interchangeable in every context.

Because Portuguese nouns have grammatical gender.

• punção is feminine → a punção
• agulha is feminine → a agulha
• hematoma is masculine → um hematoma

A useful thing to notice: nouns ending in -ma are often masculine in Portuguese when they come from Greek, such as:

• o problema
• o sistema
• o hematoma

So even though hematoma ends in -a, it is masculine.

Com a agulha means with the needle.

It is made of:

• com = with
• a = the
• agulha = needle

So literally: with the needle.

Unlike some other prepositions in Portuguese, com normally does not combine with the article a in standard modern Brazilian Portuguese. So you say:

• com a agulha
• com o médico
• com as mãos

not some contracted form.

It can feel a little redundant, yes.

Since punção already strongly suggests a puncture with a needle or similar instrument, many speakers would understand the sentence even without com a agulha:

• A punção deixou um hematoma.

Adding com a agulha can make the image more explicit or sound more clinical/technical. It is understandable, but in everyday speech people might prefer a simpler version.

Yes, deixar often means to leave, but it also has another very common meaning: to cause something to remain, to produce, or to leave behind as a result.

In this sentence:

• deixou um hematoma

means something like:

• left a bruise
• caused a hematoma
• resulted in a hematoma

So this is a very natural extended use of deixar.

Deixou is the third-person singular preterite of deixar.

That means it corresponds to:

• he/she/it left
• he/she/it caused
• it resulted in

Here the subject is A punção com a agulha, which is singular, so the verb is singular too:

• A punção ... deixou ...

If the subject were plural, the verb would change:

• As punções ... deixaram ...

Yes. Causou um hematoma is also correct.

The difference is mainly nuance:

• deixou um hematoma = left a hematoma / resulted in a hematoma
• causou um hematoma = caused a hematoma

Causou sounds a bit more direct and clinical. Deixou can sound slightly more idiomatic and natural in many contexts.

Both are good Portuguese.

Often yes, in practical meaning, but there is a nuance.

Hematoma is a medical term and can be translated as hematoma or, depending on context, bruise. In everyday Brazilian Portuguese, people often use more informal words like:

• roxo = bruise / purple mark
• sometimes mancha roxa = purple mark

So a doctor might say hematoma, while an ordinary person might say something like:

• Ficou um roxo.

It is understandable and acceptable, especially in a medical context, but it sounds a bit formal/technical.

In everyday Brazilian Portuguese, people might more naturally say:

• A agulha deixou um hematoma.
• A picada da agulha deixou um hematoma.
• A injeção deixou um hematoma.
  (if it was specifically an injection)
• Ficou um hematoma por causa da agulha.

So your sentence is fine, but somewhat more clinical.

This is a normal Portuguese sentence order:

• A punção com a agulha = subject
• deixou = verb
• um hematoma = object

Portuguese usually prefers the same basic order as English here: subject + verb + object.

The phrase com a agulha describes punção. It tells you what kind of puncture or what instrument was involved.

A rough Brazilian Portuguese guide:

• punção ≈ poon-SOWN
  The ending -ção is nasal. The n is not fully pronounced like English n.
• agulha ≈ ah-GOO-lya
  The lh sound is like the lli in million for many English speakers.
• deixou ≈ day-SHOH
  The x here sounds like sh.

A few pronunciation notes:

• ção is one of the most important Portuguese endings to learn.
• lh is a special consonant sound in Portuguese.
• In natural Brazilian speech, the vowels are often softer and more connected than in English.

Sometimes, yes, depending on the tone.

• punção = more technical/medical
• picada = more everyday, more like prick or sting

So:

• A picada da agulha deixou um hematoma.

sounds more natural in casual speech.

But picada is not always identical to punção. If you want a precise medical term, punção is better. If you want everyday speech, picada may sound more natural.