Με λίγο πείσμα μπορώ να τελειώσω αυτή τη δύσκολη άσκηση.

Greek often leaves out subject pronouns when they are not needed.

• μπορώ already means I can
• The verb ending shows 1st person singular

So Με λίγο πείσμα μπορώ να τελειώσω αυτή τη δύσκολη άσκηση naturally means I can finish this difficult exercise with a little persistence without needing εγώ.

If you did say εγώ, it would add emphasis:

• Εγώ μπορώ να τελειώσω... = I can finish...

Literally, it means with a little stubbornness or with a little persistence.

In this sentence, πείσμα sounds positive and is closer to:

• determination
• persistence
• grit

So a natural English sense is:

• With a little determination...
• If I’m a bit persistent...

με usually means with, and here it introduces the way or means by which something is done.

So:

• με λίγο πείσμα = with a little persistence
• It describes how the speaker will manage to finish the exercise

This use is very common in Greek:

• με προσπάθεια = with effort
• με υπομονή = with patience
• με τύχη = with luck

Because πείσμα is a neuter noun.

Greek words like λίγος / λίγη / λίγο change form to match the gender of the noun:

• λίγος = masculine
• λίγη = feminine
• λίγο = neuter

Since πείσμα is neuter, we get:

• λίγο πείσμα

After verbs like μπορώ (I can), Greek normally uses να before the next verb.

So:

• μπορώ να τελειώσω = I can finish

You cannot normally say:

• μπορώ τελειώσω

In Modern Greek, να is a very common particle that introduces this kind of verb form, often called the subjunctive form in grammar explanations.

This is a very important Greek pattern.

• τελειώνω = I finish / I am finishing
• τελειώσω = the form used after να here, referring to finishing the action as a whole

In μπορώ να τελειώσω, the speaker is talking about successfully completing the exercise. That is why Greek prefers the aorist-type form τελειώσω.

Very roughly:

• να τελειώσω = to finish, to get it done
• να τελειώνω = to be finishing, to finish regularly, or to focus on the ongoing process

In this sentence, the idea is a single completed result, so τελειώσω is the natural choice.

This whole phrase means this difficult exercise.

The parts are:

• αυτή = this
• τη = the
• δύσκολη = difficult
• άσκηση = exercise

The phrase is in the accusative, because it is the direct object of τελειώσω:

• finish what?
• this difficult exercise

Greek words in a noun phrase usually agree in:

• gender
• number
• case

Here they are all feminine singular accusative.

Both are possible in Modern Greek, depending on style and pronunciation.

The full form of the feminine accusative article is την, but the final -ν is often dropped before certain consonants in everyday language.

So you may see:

• τη δύσκολη άσκηση
• την δύσκολη άσκηση

Both are understandable, and τη δύσκολη is very common.

The same thing happens with the demonstrative sometimes:

• αυτή τη is common
• αυτήν τη is also possible in more careful or formal usage

No. πείσμα can be either:

• negative: stubbornness, pigheadedness
• positive: determination, persistence

The exact meaning depends on context.

In this sentence, it clearly has a positive sense, because it helps the speaker succeed:

• Με λίγο πείσμα... = With a little persistence...

So here it is much closer to determination than to mere stubbornness.

Yes. Greek word order is more flexible than English word order.

The sentence as given is natural and neutral:

• Με λίγο πείσμα μπορώ να τελειώσω αυτή τη δύσκολη άσκηση.

But other orders are also possible, with slightly different emphasis, for example:

• Μπορώ να τελειώσω αυτή τη δύσκολη άσκηση με λίγο πείσμα.
• Αυτή τη δύσκολη άσκηση μπορώ να τη τελειώσω με λίγο πείσμα.

The original version puts Με λίγο πείσμα first, which gives a little emphasis to the idea of persistence as the key to success.

Not always.

άσκηση can mean:

• an exercise in a book or class
• practice
• physical exercise
• a drill or training exercise

In this sentence, because of δύσκολη and the verb τελειώσω, the most natural meaning is:

• exercise in the sense of a task, problem, or assignment

So here it most likely refers to a school/book exercise or some other difficult task to complete.