Po treningu boli mnie nadgarstek i mam małą ranę na ręce.

Because po meaning after takes the locative case in Polish.

• dictionary form: trening
• locative singular: treningu

So po treningu means after training / after the workout.

You will see the same pattern in:

• po pracy = after work
• po obiedzie = after lunch

This is the normal Polish pattern with boleć (to hurt).

Polish structures it like this:

• nadgarstek = the thing that hurts
• boli = hurts
• mnie = me

So the literal structure is the wrist hurts me, not I hurt wrist.

This is very natural in Polish:

• Boli mnie głowa = My head hurts
• Bolą mnie plecy = My back hurts

Because Polish often leaves out possessives with body parts when the owner is obvious.

In boli mnie nadgarstek, the word mnie already shows who is affected, so mój is unnecessary.

Likewise, in mam małą ranę na ręce, it is naturally understood to be my hand/arm unless context says otherwise.

You can add mój for emphasis, but it is not the neutral default:

• Boli mnie mój nadgarstek sounds marked or emphatic.

As a learner, it is safest to use boli mnie nadgarstek.

With boleć, the standard pattern is:

• boli mnie
• boli cię
• boli go
• boli ją

You may hear boli mi... in some colloquial speech, but boli mnie... is the standard form you should learn first.

Because after mam (I have), the direct object goes into the accusative case.

Dictionary form:

• mała rana = a small wound

After mam:

• mam małą ranę

Both words change because they must agree in case:

• mała → małą
• rana → ranę

This is a very common feminine singular pattern in Polish.

It is the locative singular of ręka.

That is because na can take different cases depending on meaning:

• na + locative = location, on
• na + accusative = movement onto, onto

Here the wound is located on the hand/arm, so Polish uses na ręce.

Compare:

• rana na ręce = a wound on the hand/arm
• założyć coś na rękę = to put something onto the hand/arm

Because the wound is on the surface of the body.

Polish normally uses na for things located on the outside of a body part:

• rana na ręce
• tatuaż na ramieniu
• blizna na nodze

w ręce usually means something different, such as:

• in the hand
• inside the arm/hand
• sometimes in someone’s possession

So na ręce is the natural choice here.

It can mean either, depending on context.

That is one difference from English: Polish ręka is broader.

If you want to be more specific:

• dłoń = hand / palm
• palec = finger
• nadgarstek = wrist
• przedramię = forearm
• ramię = upper arm / arm

So rana na ręce can mean a wound on the hand or arm unless the context makes it clear.

Because the grammatical subject is singular: nadgarstek.

• nadgarstek boli = the wrist hurts
• therefore: boli mnie nadgarstek

If the thing hurting were plural, the verb would also be plural:

• Bolą mnie ręce = My hands/arms hurt
• Bolą mnie nadgarstki = My wrists hurt

No, Polish word order is fairly flexible, but this sentence has a very natural neutral order.

Po treningu at the beginning sets the time frame first:

• After training, ...

Then the sentence gives the two problems:

• boli mnie nadgarstek
• i mam małą ranę na ręce

You can move things around for emphasis, but this version is a good standard model for learners.