I think a stage has been missed here and if not addressed it will come back to bite you as a pretty straightforward challenge. People very rarely just go into care. More often than not there is a trial period and it’s a genuine one too i.e. if the care doesn’t work or suit then either other locations or types of accommodation are tried or people go home with comprehensive packages of care etc.
The sons view of his mothers intent is nothing to the point. Ditto a social work perspective, which often amounts to little more than “Well it’s temporary with a view to permanent but really she won’t be coming home” when they often have no insight into whether that’s how things will pan out at all.
She moved in on 5/5/21. At the point she moved in was that temporary with a view to permanent? If so then your end date for HB and the amount of the o/p will both be wrong.
