Is there a word beyond reflection, that is 'using the past mistakes to learn for the future?'

If memory serves me, "reflection" does not necessarily mean drawing from the past. You might consider "self-correction (or self-correcting)" to describe learning from one's mistakes. Words along the line of being reminded (remembrance), could be "retrospection", "recollection", or "hindsight" as possible noun choices for "reflection". They have verb/adj/idiomatic forms as well. Their meanings all reflect the idea of drawing on past knowledge for present application (not necessarily past mistakes). From these, you will probably find other possibilities.

https://www.thefreedictionary.com/self-correction

https://www.thefreedictionary.com/reflection

https://www.thefreedictionary.com/retrospection

https://www.thefreedictionary.com/recollection

https://www.thefreedictionary.com/hindsight