I completed the event that rewards the bajorian interceptor t6 ship with the bajor warp/singularity cores to unlock the rewards across my user account (during event run). However, I am unable to claim the bajor singularity core on my human federation character who has access to a Romulan flagship t6 ship. It is available for my Romulan captains, but not non-Romulan characters.
Bajor Defense Hyper Injection Singularity Core and Kobali field stabilizing singularity core both do not grant the ability to reclaim the cores on non-Romulan captains.
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There is an achived Reddit thread asking the same question, one answer seems to be you can only claim 1 if you already have the warp core you cannot reclaim the singularity as well, but some agree that fed captains don't get the option. Contact the Support crew. Anyway here is the link
https://www.reddit.com/r/sto/comments/cnqalx/bajor_defense_hyper_injection_singularity_core/