You do not need to send any proof of funds received to anyone. HMRC will only need proof (ie bank statements etc) if they decide to audit you. Being audited is a very serious, but pretty unusual thing. Ultimately, as long as you are truthful about the amounts you are earning, you would have nothing to fear from an audit.
I would suggest setting up a simple Excel spreadsheet that looks something like this:
DATE | DESCRIPTION | AMOUNT
14.04.16 | Joe Bloggs tuition | £35
I'm afraid the formatting of this message board doesn't let me type the above very clearly, but you get the idea.
Fill it in as you go along, and then when you come to having to fill in your tax return, it will be a very simple job to calculate the total money you've earned in a year. The tax return only asks you for the sum total amount you've earned - you do NOT need to provide them with a breakdown. The spreadsheet is really just for you to keep track of it all, and to make things simple if you were ever audited. It almost seems too simple, I know, but that's really all you need in this kind of situation.
In terms of allowable expenses, it doesn't have to be complicated. You can go into great detail and start claiming for percentages of the cost of heating your home (if you're working from home) but a lot of people feel that's more trouble than it's worth. Generally, the kinds of expenses that you might want to offset against your income might be things like text books and other materials that are clearly only for your tutoring. Set up another part of your spreadsheet to keep a record of these things:
DATE | DESCRIPTION | AMOUNT
4.12.15 | Subscription to online Maths revision website | £35
10.02.16 | WHSmith Notebooks | £5.50
13.04.16 | OCR GCSE Maths textbook | £20
Again, HMRC will only ask you for a total £number when it's time to fill out your tax return. You do not need to show the breakdown, but you should keep any receipts (again, just in case you are ever audited).
There's also a lot more detail about all of this stuff in our free e-book, which you can download here:
http://www.thetutorpages.com/free-tutoring-ebook