If I use Exeggcute's "Propagation" Ability to retrieve it from the Discard Pile into my hand, then use something like Ultra Ball or Computer Search to discard Exeggcute, would I be able to use "Propagation" again to retrieve it from the Discard Pile?

Yes, whenever Exeggcute is in the Discard Pile, you may use Propagation during your turn as long as its Ability has not been disabled.

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