Can dua change my destiny?

What the Prophet ﷺ said

The Prophet ﷺ said: "Nothing repels the decree except dua, and nothing increases life span except righteousness" (Jami' at-Tirmidhi 2139; graded hasan by al-Albani, though some scholars grade it weak). So dua isn't a formality on top of a fixed script. It's one of the means Allah has placed in the world, just like medicine for illness or work for income.

How does that fit with qadr?

Muslims believe Allah knows and has written everything. That sounds like dua should change nothing, but scholars answer it like this:

• The dua is part of the decree. Allah decreed that you would be healed because you asked. The cause and the result were written together.
• Some things are written as conditional in what angels record, for example "if he prays, this happens". Many scholars link this to Quran 13:39: "Allah eliminates what He wills or confirms, and with Him is the Mother of the Book." The final, perfect knowledge with Allah never changes.
• You don't know what's written, so you're never stuck. Your job is to ask, work and trust.

For more on this, see What is qadr in Islam?

Is every dua answered?

Scholars teach, based on the hadith, that a sincere dua is always answered in one of three ways: Allah gives you what you asked for, keeps a harm away from you of the same size, or saves the reward for you in the Hereafter. So a dua is never wasted, even when the result looks different from what you pictured. See Why isn't Allah answering my dua?

Keep asking

Because dua can change things, it makes sense to ask for big things too: a righteous spouse, healing, guidance for family. Keep going even if it takes years. See Is it okay to keep making the same dua for years? At the same time, take the practical steps; dua and effort go hand in hand.