Yes they will accept records not completed by an Accountant.
Send them what you send to HMRC.
I was self-employed for many years whilst claiming Carer’s Allowance. And used to send in my “Accounts” once a year.
There may be a form to complete which asks for more detail regarding some of the expenses but they didn’t always ask for that information. It was mainly to certify that all the expenses were 100% business related. If you have claimed for things like telephone expenses they wanted to know what proportion was private use.
When you complete your Self Assessment for HMRC you must include the Carer’s Allowance you have received in that tax year because it is taxable income.
Make sure that your profits are below the threshold for Carer’s Allowance. Currently in the 2022/23 tax year, if you earn £132 per week or more you will not be entitled to Carer’s Allowance. I believe it is averaged out for the year when you are self employed.