Thanks for asking! During war, only combat activities (faction bonds & bounties) cause positive influence shift. The easiest way to help is by cashing SINC bounties in systems that we control. It's important to note that E:D cares about the number of transactions, as opposed to the value of bounties. In other words: frequent, small bounty redemptions are best.
Bonds only have effect in system that's at War. You can cash them elsewhere, but I don't think it helps. Generally speaking, it's easier to get bounties than combat bonds. Occasionally, we trigger Wars that we'd rather not win. When this happens, we may need to abstain from combat or fight against our own faction. It depends on the balance of power for a given day.