It's only your husband who cannot claim public funds. You can claim anything you are entitled to, because you are a UK citizen.
The only thing you can't do is claim MORE because of your husband. For example, you can only claim the benefit allowance for a single person and not a couple, or you can only qualify for the size of council house allowed for a single person and not a couple.
So, if you are a carer and you are being offered support/help with your carer duties, then that comes under your benefit entitlement, not your husband.
What you can't do is claim any kind of disability allowance, or support for your husband, so as long as any support is for you and is not in his name, I don't see why it would affect his immigration status.
Do you have a webpage link or anything to the service that's being offered?