Hi,
Welcome to Benefits and Work
If your forum name is also your real name, you may wish to change it to maintain some anonymity.
This can be done quite easily by following the instructions on this FAQ :
My full name is showing, how can I stop it? For entitlement to a Cold Weather Payment one of the qualifying criteria is that you would have to be in receipt of Income Related (IR) ESA.
You state that you receive Contribution Based (CB) ESA, but do you also receive any IR ESA top-ups in the form of the "Disability Income Guarantee".
You should be able to clarify this in your ESA benefit breakdown letter.
This FAQ should help to explain it for you :
Cold Weather Payments bro58