Not necessarily - although payment may be made to a third party, it is still the tenant's Rent Supplement claim, and it would not be appropriate to give details of their claim to another person. If there have been a change in circumstances (e.g. tenant no longer qualifies for Rent Supplement), it is his/her responsibility to inform the landlord, not the HSE.
If the tenant received their payslip (the weekly letter referred to by the OP), I wonder if the delay in the payment was caused elsewhere? The payslip is issued by the same computer system as the payment itself, and both issue on the same night - perhaps there was a delay at the bank which was outside the control of the CWO/HSE?