Do you keep the rewards already earned and chosen?
If so, do you end up with an extra 4 rewards?
No.
If not, do you get the points back?
Yes.
Is the secondary aspiration affected in any way?
Do you get the option to change that as well?
Secondary aspiration is a reward perk, see above.
If not, does it stay untouched, get overwritten by the old main aspiration or disappear?
If it goes, do you lose the rewards &/or points?
Your points and bar stay as they are, although any perk points allocated are unallocated and must be reallocated. Simple, no pain.