I am looking for the German equivalent of the phrase "make a difference", like in the sentence "In my job, I want to do something that really makes a difference".
Would "etwas zu bewegen" be a good choice?
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Another alternative could be:
As for example in:
Ein Bereich, in dem der EU-Haushalt etwas bewirken kann und sollte, ist ganz klar Forschung und Technologie.
An obvious area where the EU budget can, and should, make a difference is research and technology.
Your proposal is indeed a good choice (»Ich will etwas bewegen«). Alternatively you could use:
Ich will die Welt [ein bisschen] besser machen / verbessern.
Another alternative translation would be
einen Wandel herbeiführen
if you want to stick with as much as possible German equivalent of the make a difference.