Train up the Diplomacy skill, that should hopefully raise your negative Faction standings up enough so you can safely travel through those systems.
If Diplomacy is already at level 5 and the negative Faction is at or below -5.00 standing then you’ll have to do a little bit of mission running. It shouldn’t take a lot with Social and Connections skills trained up.
Anyway, check out the ‘Faction Standing Repair Plan’ for info on raising your standings with Agents that give Faction standing increase.