Tax Credits - New Zealand reciprocal agreement?

British Citizen claimed WTC (inc dis element) then became a couple with a New Zealand National (currently on a FLR(M) visa) with no recourse. Reported to tax credits, partner added to claim - income counted but no second adult element. Seems right so far.
They have a baby in 2019, reported to TC office, appropriate CTC elements added.
In Dec 2020, amended award notice received to now include a second adult element, over £1000 backdated tax credits received. NZ National has now resided in UK for 5 years but still on the same visa, which is due for renewal soon. They have queried several times with TC office but they’re not very reassured that they are entitled to the second adult element and have concerns that this will affect visa renewal.

It looks like there is a reciprocal agreement in place for Child Benefit for NZ Nationals living in the UK, but I can’t find anything in place for any other benefits. Not sure if I’m on the right path or not.

Thanks :)

Usual disclaimer about licensed immigration advisers ... not providing advice, just commenting on the law as a matter of academic interest ... blah blah blah.  But:

For the purpose of these rules,

....

(ii) subject to subparagraph (iii) below, a person (P) shall not be regarded as having recourse to public funds if P is entitled to benefits specified under section 115 of the Immigration and Asylum Act 1999 by virtue of regulations made under sub-sections (3) and (4) of that section or section 42 of the Tax Credits Act 2002;

https://www.gov.uk/guidance/immigration-rules/immigration-rules-introduction