treatment of UC arrears

Any help please on this one?
Cl and ptr form a relationship in March ‘19. . Ptr is working, there are 3 children and housing costs. Cl is deemed, incorrectly, to have no recourse.  When I find this, they put entries on the journal, we send in paper MR in August (never gets there).  I make a number of requests to the local partnership, but we are consistently told she has no recourse.
In December, I go in person to local Jobcentre, to take in a duplicate MR paper form.  Officer there looks into the case: in my presence and sees that the client was found to have recourse in May 2019 on a revision, but they’d not changed the front household screen and had never paid the correct award for a couple.  Clt had a payment of arrears just before Christmas of something over £1300.
The LA, who had to comply with whatever UC say, had processed CTR with a single person’s and children’s allowances.
We asked them to revise the household screens to include the couple’s allowance.
They’ve done so, but seem to have also apportioned the UC lump sum to the CTR calculation month by month, generating a new CTax bill of nearly £300..

My question: 

is it correct for the LA to add in the new UC awards month by month (as in the old relevant benefit HB rules)?

or should the LA have just changed their own household screens and counted the UC underpayment as a single lump sum payment (an in the old treatment of TC arrears/one off payments)?

(I thought I’d read somewhere that such lump sum payments were ignored.  But was this just really wishful thinking? - I can’t find anything on the forum.  Or should I be looking in the LA’s own CTR scheme?  Or seek compensation for maladministration?)

Thanks everyone, as always