Not much i'm afraid, you can't claim JSA (jobseekers allowance) as you are working 16 hrs pw, if you were doing under 16 hrs and looking for more hours you could then claim JSA.
you can't claim Incapacity benefit unless you are off sick from your employer and even then you may not qualify as it is based on national insurance contributions and if you are working 16 hrs i doubt you are paying any.
it would be worth your partner enquiring to the jobcentre about claiming Income support as a top up to his incap but this is a mean tested benefit and household income/saving etc are taken into account when calculating entitlement.