Is there a way to access the cores of hexes fully separated from the main sprawl by empty water hexes? It is especially problematic on Fulgora.

If your center island of hexes doesn't connect to all Fulgora hexes, then you must have gotten pretty unlucky. I set the default mod settings which controls its connectivity to a value that seemed to produce full connectivity on every Fulgora map generation. That is unfortunately the consequence of only relying on randomness to connect all hexes, risking the possibility of producing a disconnected set of hexes. You are at least able to run the /claim command to claim the hex you're standing/flying over for free. That works on hexes that don't have hex cores to be claimed normally. Doing this then lets you claim more hexes normally.

Thanks!
Then I must be really unlucky, because I have multiple unconnected larger islands close to my center. I am using the default map settings.
https://freeimage.host/i/FPwSpQS

The claim command seems to only work on hexes where the core is already visible.

Oh, oops. I forgot it was supposed to be the /force-claim command, but I just tested that and for some reason it does not claim liquid hexes. I must've somehow broken that with some update because I remember it working as expected in the past. Sorry, it seems there might be no way for you to get across the heavy oil ocean until I release the next update. Been putting that off for a bit, I'll be getting back into soon enough.