This guy does not have a faction or policy issue, he is at net -5 base influence from a regular empire. He's got 0.5 from a rivalry and 1.15 from factions, which is entirely reasonable non-negative number, so being negative is very strange. He should be at +4.65.
The only thing I can think of that costs -5 a month is Subject Integration and I feel like that should be in the tool tip. Otherwise... losing multiple humiliation wars? There's something strange here and only OP can tell us if this is an unmodded game.
Can't be integration cuz it would be listed as -5 integration.
The only thing I can think of that reduce base influence gain is from faction management (suppress or promote).
*EDiT*
I just realized you misunderstood OP screenshot, that's why you think op have 4.65 income thus must have -5 negative income some where.
Here is what you are missing OP doesn't have 2 base income but -2 and only way for base influence to be negative is what I said, faction suppression or promotion.
Meaning -2 from base, +1.14 from faction and +0.5 from rival, sum to -0.36 which is pretty close to -0.35 net influence OP has.
In conclusion check your faction, do you have any faction that you set them to suppress or promote and forgot about them?