I think you will need to clarify a few things.
Is the carer supplied by and paid by the council?
You say "paid via direct payments from my local council", is this a 'Direct Care Payment' arrangement?
Is the carer working for an agency, or working alone?
So:
Is the carer getting paid by the council?
Are you getting direct payment from the council and using it to pay an agency?
Or are you getting the direct payment from the council and using it to pay the carer?
If the carer is paid by the council then they are employed by the council, and this is not a direct care arrangement.
(The carer should be getting a payslip from the council).
If you are getting direct payment and using it to pay an agency for care then the agency is the carers employer, and so is responsible for their holiday pay, etc.
(The carer should be getting a payslip from the agency).
If you are getting the direct payment and using it to pay an individual then you are the carers employer - and so you are responsible for paying holiday pay, statuatory sick pay, National Insurance, Income Tax, etc. just like any other employer.
(The carer should be getting a payslip from you).
So you would be responsible for paying the carers statuatory holiday pay, out of your Direct Payment.
Using direct payments to employ a carer
The exception to this would be if the carer is working as self employed, usually this would mean they care for more than one person but are not working for an agency.
If you are employing your carer as an individual then you realy need to check out your legal responsibilities as an employer.
Your council should have made you aware of this when you were awarded direct payments, councils often have a dedicated team who can help with your responsibilities as the employer of your carer.
You may find this interesting/helpful.
Disabled and elderly people and their personal assistants
General information about Direct Payments for care.
https://www.moneyadviceservice.org.u...aying-for-care