Benefit claims are normally processed on a computer system but sometimes that is not possible either because of a system fault or if there are peculiarities about the case. If it can't be processed on the computer it's done clerically and that's why you've been telt it's clerical.
If I were you I'd get in touch wi the call centre and tell them you want an urgent call back from the processing section within an hour to speak to about the case. They may say it's a three hour call back, if you're okay wi three hours rather than an hour fine, if not tell them it's causing you hardship and you want an urgent call back.
When you speak to the call centre tell them you have a copy of the letter telling you the Carer's Allowance has stopped and you want to post or fax it to the ESA section (if someone can go to the local Jobcentre for you ask if the Jobcentre will fax it for you. You are best to get the name of someone to fax it to so you can chase it up wi that person if need be.)
The call centre can notify the appropriate section and you should get a call back from the processing section within the agreed time. If they don't call back within the time, call again tell them you've not had a call back and it then gets escalated to higher levels and people in higher grades start getting involved.
If you put in an application for a Crisis Loan, the Social Fund section should work out there are arrears of ESA due and they will refuse to pay because the ESA section should pay the arrears. They may or may not get in touch wi the ESA section and insist that the ESA section pay the arrears asap to avoid the crisis.