Personally I would not pay for your partner's CSA liabilities.
It all depends on whether your partner is on the CSA 'old' scheme or the 'new' scheme, he should know this from his CSA correspondence.
If he is on the old scheme (CSA 1) then the 'non-resident' parent partner's (NRPP) income can be taken into account. CSA 1 scheme is very complicated to calculate - which is why they nearly always got the early cases wrong by thousands and people topped themselves - it takes into account housing costs, living costs (council tax etc).
The new scheme (CSA 2) is much simpler and operates like a "dad tax" whereby CSA assessment is 15% net income for 1 child, 20% for 2 etc, with reductions such as 1/7th reduction if child stays for overnight staying contact for more than 52 nights a year, 2/7th reduction for more than 104 (I think) etc.
If you are on CSA2 scheme then they may ask about your parnter's details/income but YOU ARE UNDER NO OBLIGATION TO DO THIS. They may threaten to put you on a Interim Maintenance Assessment (which is about £30 pw).
I would NEVER correspond with the CSA by telephone - they frequently get it wrong and lie. Always communicate in writing that way there is evidence.
As you are not going to earning any money you will be nil-assessed (£5 per week). As long as you satisfy the csa that you aren't doing a dodge.
I would recommend joining NACSA for the best advice and support,
I've recently been through the CSA, luckily I have a very well defined shared residency order (so even though the law says I am an equal legal parent, as the ex is in receipt of the child benefit, she is defined as the parent with care and I'm suddenly the non-res parent!). My daughter lives with me for approx 35% of the year, I have a partner and new baby, I was paying the ex under a private arrangement by standing order - she thought she could get more money, so involved the CSA, now she gets less! crazy! I provide for my daughter when she is in my care - clothes, food, bedroom etc.